9466:
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9461:{\displaystyle {\begin{alignedat}{6}a\cdot (\mathbf {v} \otimes \mathbf {w} )~&=~(a\cdot \mathbf {v} )\otimes \mathbf {w} ~=~\mathbf {v} \otimes (a\cdot \mathbf {w} ),&&~~{\text{ where }}a{\text{ is a scalar}}\\(\mathbf {v} _{1}+\mathbf {v} _{2})\otimes \mathbf {w} ~&=~\mathbf {v} _{1}\otimes \mathbf {w} +\mathbf {v} _{2}\otimes \mathbf {w} &&\\\mathbf {v} \otimes (\mathbf {w} _{1}+\mathbf {w} _{2})~&=~\mathbf {v} \otimes \mathbf {w} _{1}+\mathbf {v} \otimes \mathbf {w} _{2}.&&\\\end{alignedat}}}
6266:
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as the unit "balls" enclose each other, a sequence converges to zero in one norm if and only if it so does in the other norm. In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector
187:, which, roughly speaking, specifies the number of independent directions in the space. This means that, for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure are exactly the same (technically the vector spaces are
9110:
14136:
can be interpreted in terms of
Hilbert spaces. For example, a great many fields in physics and engineering lead to such equations, and frequently solutions with particular physical properties are used as basis functions, often orthogonal. As an example from physics, the time-dependent
10563:
2839:. In the special case of two arrows on the same line, their sum is the arrow on this line whose length is the sum or the difference of the lengths, depending on whether the arrows have the same direction. Another operation that can be done with arrows is scaling: given any positive
3207:
3991:
12465:
10370:
7697:
2662:
was the first to give the modern definition of vector spaces and linear maps in 1888, although he called them "linear systems". Peano's axiomatization allowed for vector spaces with infinite dimension, but Peano did not develop that theory further. In 1897,
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5515:
4723:
12351:
13522:
11016:
10775:
15122:
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studies the isomorphism classes of all vector bundles over some topological space. In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real
13148:
9002:
1727:
14057:, and is much applied in engineering. More generally, and more conceptually, the theorem yields a simple description of what "basic functions", or, in abstract Hilbert spaces, what basic vectors suffice to generate a Hilbert space
12991:
12631:
5371:. It is an isomorphism, by its very definition. Therefore, two vector spaces over a given field are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space over a given field is
11359:
10458:
13880:
10220:
9812:
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14386:: using that the sum of two polynomials is a polynomial, they form a vector space; they form an algebra since the product of two polynomials is again a polynomial. Rings of polynomials (in several variables) and their
12200:
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are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property. Therefore, the set of such functions are vector spaces, whose study belongs to
1227:
11864:
3008:
11870:—focuses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence. The image at the right shows the equivalence of the
9826:
From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space over a given field is characterized, up to isomorphism, by its dimension. However, vector spaces
8929:
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6166:
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under addition, and requiring that scalar multiplication commute with the tensor product ⊗, much the same way as with the tensor product of two vector spaces introduced in the above section on
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if and only if they have the same coefficients. Also equivalently, they are linearly independent if a linear combination results in the zero vector if and only if all its coefficients are zero.
13339:
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3533:. The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic. The example of complex numbers is essentially the same as (that is, it is
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An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms (related to vector addition) say that a vector space is an
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1981:. A linear subspace is a vector space for the induced addition and scalar multiplication; this means that the closure property implies that the axioms of a vector space are satisfied.
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4107:{\displaystyle {\begin{alignedat}{9}&&a\,&&+\,3b\,&\,+&\,&c&\,=0\\4&&a\,&&+\,2b\,&\,+&\,2&c&\,=0\\\end{alignedat}}}
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is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero. The linear transformation of
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are a fundamental tool for the study of vector spaces, especially when the dimension is finite. In the infinite-dimensional case, the existence of infinite bases, often called
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From a conceptual point of view, all notions related to topological vector spaces should match the topology. For example, instead of considering all linear maps (also called
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Der
Barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behandlung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of geometry)
14326:
10449:
8954:
8860:
4411:
19479:
18493:"Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (On operations in abstract sets and their application to integral equations)"
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of
Lebesgue measure, being unbounded, cannot be integrated with the classical Riemann integral. So spaces of Riemann integrable functions would not be complete in the
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Banach and
Hilbert spaces are complete topological vector spaces whose topologies are given, respectively, by a norm and an inner product. Their study—a key piece of
7186:
5738:{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n})\mapsto \left(\sum _{j=1}^{n}a_{1j}x_{j},\sum _{j=1}^{n}a_{2j}x_{j},\ldots ,\sum _{j=1}^{n}a_{mj}x_{j}\right),}
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4852:{\displaystyle {\begin{aligned}f(\mathbf {v} +\mathbf {w} )&=f(\mathbf {v} )+f(\mathbf {w} ),\\f(a\cdot \mathbf {v} )&=a\cdot f(\mathbf {v} )\end{aligned}}}
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complete, which may be seen as a justification for
Lebesgue's integration theory.) Concretely this means that for any sequence of Lebesgue-integrable functions
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of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly define objects by specifying maps from or to this object.
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In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones.
16279:
A basis of a
Hilbert space is not the same thing as a basis of a linear algebra. For distinction, a linear algebra basis for a Hilbert space is called a
11635:. Roughly, a vector space is complete provided that it contains all necessary limits. For example, the vector space of polynomials on the unit interval
4213:
They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too.
2654:
are present. Grassmann's 1844 work exceeds the framework of vector spaces as well since his considering multiplication led him to what are today called
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10164:
2516:
between vectors and their coordinate vectors maps vector addition to vector addition and scalar multiplication to scalar multiplication. It is thus a
15915:
It is also common, especially in higher mathematics, to not use any typographical method for distinguishing vectors from other mathematical objects.
19111:
9105:{\displaystyle \mathbf {v} _{1}\otimes \mathbf {w} _{1}+\mathbf {v} _{2}\otimes \mathbf {w} _{2}+\cdots +\mathbf {v} _{n}\otimes \mathbf {w} _{n},}
7290:
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studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations. In his work, the concepts of
2143:
12122:
8169:
18750:
17854:
601:
15501:, yield modules. The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have
10558:{\displaystyle \mathbf {x} \cdot \mathbf {y} =\cos \left(\angle (\mathbf {x} ,\mathbf {y} )\right)\cdot |\mathbf {x} |\cdot |\mathbf {y} |.}
13737:
The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red).
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if its elements are linearly independent and span the vector space. Every vector space has at least one basis, or many in general (see
8887:
6870:
6930:
1784:
18115:, The Wadsworth & Brooks/Cole Mathematics Series, Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software,
8226:
and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements for
19622:
19284:
10882:. Note that in other conventions time is often written as the first, or "zeroeth" component so that the Lorentz product is written
6168:
By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in
14117:. Not only does the theorem exhibit suitable basis functions as sufficient for approximation purposes, but also together with the
10568:
4222:
19955:
19520:
15747:
14086:
of their span (that is, finite linear combinations and limits of those) is the whole space. Such a set of functions is called a
10160:
respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm
10123:
5468:
are a useful notion to encode linear maps. They are written as a rectangular array of scalars as in the image at the right. Any
3253:
20013:
16506:
14999:
10784:
3235:
itself with its addition viewed as vector addition and its multiplication viewed as scalar multiplication. More generally, all
3202:{\displaystyle {\begin{aligned}(x_{1},y_{1})+(x_{2},y_{2})&=(x_{1}+x_{2},y_{1}+y_{2}),\\a(x,y)&=(ax,ay).\end{aligned}}}
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norm, and the orthogonal decomposition would not apply to them. This shows one of the advantages of
Lebesgue integration.",
8339:
19154:
6996:
18680:
13986:
with desirable properties that approximate a given limit function is equally crucial. Early analysis, in the guise of the
16395:
16044:
13959:
By definition, in a
Hilbert space, any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions
3852:
2520:, which allows translating reasonings and computations on vectors into reasonings and computations on their coordinates.
2109:
16586:
12460:{\displaystyle \|\mathbf {x} \|_{p}:=\left(\sum _{i}|x_{i}|^{p}\right)^{\frac {1}{p}}\qquad {\text{ for }}p<\infty .}
18394:
13279:
10365:{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =\mathbf {x} \cdot \mathbf {y} =x_{1}y_{1}+\cdots +x_{n}y_{n}.}
8265:
7402:
3805:
also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions
1154:
under addition, and the four remaining axioms (related to the scalar multiplication) say that this operation defines a
594:
9582:
3717:
3566:
then we see that the rules for addition and scalar multiplication correspond exactly to those in the earlier example.
11707:
6442:
are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set
2825:
spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the
1368:
18534:
Betrachtungen über einige
Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry)
15936:
a scalar that produces a vector, while the scalar product is a multiplication of two vectors that produces a scalar.
9941:
20064:
20003:
7692:{\displaystyle a_{0}f+a_{1}{\frac {df}{dx}}+a_{2}{\frac {d^{2}f}{dx^{2}}}+\cdots +a_{n}{\frac {d^{n}f}{dx^{n}}}=0,}
5134:
For example, the arrows in the plane and the ordered pairs of numbers vector spaces in the introduction above (see
176:, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying
99:
15381:
Properties of certain vector bundles provide information about the underlying topological space. For example, the
14331:
General vector spaces do not possess a multiplication between vectors. A vector space equipped with an additional
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17954:
15194:
11671:
10778:
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8607:
1331:
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11627:
A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where any
7477:
3613:
1476:
1417:
18949:
18737:
Calcolo
Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva
18421:
16734:
15784:
13344:
12998:
11170:
7882:
16475:
15335:. Despite their locally trivial character, vector bundles may (depending on the shape of the underlying space
1732:
19593:
18976:
18874:
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18438:
18156:
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of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the
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14014:
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587:
16538:
4553:
20054:
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19615:
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with the same topology is complete. A norm gives rise to a topology by defining that a sequence of vectors
8069:
5974:
4691:
3975:
224:
17:
14125:. Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional
12062:
is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.
8491:
19848:
19698:
19588:
19423:
18066:
16766:
15339:) be "twisted" in the large (that is, the bundle need not be (globally isomorphic to) the trivial bundle
14407:, which are neither commutative nor associative, but the failure to be so is limited by the constraints (
13228:
12346:{\displaystyle \|\mathbf {x} \|_{\infty }:=\sup _{i}|x_{i}|\qquad {\text{ for }}p=\infty ,{\text{ and }}}
11368:
3985:
3979:
3537:
to) the vector space of ordered pairs of real numbers mentioned above: if we think of the complex number
2556:
introduced, in 1804, certain operations on points, lines, and planes, which are predecessors of vectors.
2385:
1843:
456:
177:
13885:
13556:
8538:
6835:
6800:
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2667:
adopted Peano's axioms and made initial inroads into the theory of infinite-dimensional vector spaces.
2335:
1260:
19753:
19647:
18134:
15602:
14759:
13517:{\displaystyle \lim _{k,\ n\to \infty }\int _{\Omega }\left|f_{k}(x)-f_{n}(x)\right|^{p}\,{d\mu (x)}=0}
12011:
11913:
11011:{\displaystyle \langle \mathbf {x} |\mathbf {y} \rangle =-x_{0}y_{0}+x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}.}
10375:
9831:
do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions
9640:
8485:. Addition and scalar multiplication is performed componentwise. A variant of this construction is the
8256:
of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space.
8155:
6196:
5962:
2620:
2226:
211:
and related areas. Infinite-dimensional vector spaces occur in many areas of mathematics. For example,
18690:
11749:
11516:
10770:{\displaystyle \langle \mathbf {x} |\mathbf {y} \rangle =x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}-x_{4}y_{4}.}
10618:
9900:
8727:
8432:
7084:
7055:
2112:). This is a fundamental property of vector spaces, which is detailed in the remainder of the section.
19993:
19642:
13752:
8618:
8159:
8123:
7507:, which is precisely the set of solutions to the system of homogeneous linear equations belonging to
6175:
5286:
2715:
2513:
1986:
19499:
16237:
11200:
10090:
19985:
19868:
18061:
15597:
are vector spaces whose origins are not specified. More precisely, an affine space is a set with a
14306:
14118:
11947:
11245:
also has to carry a topology in this context; a common choice is the reals or the complex numbers.
11026:
10429:
8996:
as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called
8934:
8840:
8163:
4391:
2517:
2087:
251:
184:
19583:
15117:{\displaystyle \mathbf {v} _{1}\otimes \mathbf {v} _{2}=-\mathbf {v} _{2}\otimes \mathbf {v} _{1}}
12039:
11778:
11088:
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7223:
6758:
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6335:
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3005:. The sum of two such pairs and the multiplication of a pair with a number is defined as follows:
20031:
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19608:
18500:
15886:
15598:
15307:
14204:
14114:
13991:
12059:
11632:
11056:
8654:
6075:
3574:
2573:
2502:
1126:. These two cases are the most common ones, but vector spaces with scalars in an arbitrary field
1025:
547:
161:
38:
14914:{\displaystyle \mathbf {v} _{1}\otimes \mathbf {v} _{2}\otimes \cdots \otimes \mathbf {v} _{n},}
14138:
14049:
can be approximated as closely as desired by a polynomial. A similar approximation technique by
13702:{\displaystyle \lim _{k\to \infty }\int _{\Omega }\left|f(x)-f_{k}(x)\right|^{p}\,{d\mu (x)}=0.}
12720:
9855:
9517:
5326:
are completely determined by specifying the images of the basis vectors, because any element of
4720:
that reflect the vector space structure, that is, they preserve sums and scalar multiplication:
3923:
and similarly for multiplication. Such function spaces occur in many geometric situations, when
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231:
165:
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16760:
16744:
16564:
16548:
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16405:
16209:
16122:. For technical reasons, in the context of functions one has to identify functions that agree
15672:{\displaystyle V\times V\to W,\;(\mathbf {v} ,\mathbf {a} )\mapsto \mathbf {a} +\mathbf {v} .}
11603:
9938:, which relies on the ability to express a function as a difference of two positive functions
7191:
4351:
4176:
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19810:
19805:
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15744:
An important example is the space of solutions of a system of inhomogeneous linear equations
14702:
14544:
14158:
14133:
13162:
13143:{\displaystyle \|f\|_{p}:=\left(\int _{\Omega }|f(x)|^{p}\,{d\mu (x)}\right)^{\frac {1}{p}}.}
11953:
11893:
11496:
8243:
7159:
6260:
6220:
5770:
3212:
1958:
718:
111:
18625:
12663:
11568:
8075:
4217:
can be used to condense multiple linear equations as above into one vector equation, namely
4139:
1234:
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534:
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parametrized by the points of a differentiable manifold. The tangent bundle of the circle
14356:
13527:
12749:
8150:, that is, a corpus of mathematical objects and structure-preserving maps between them (a
1722:{\displaystyle a_{1}\mathbf {g} _{1}+a_{2}\mathbf {g} _{2}+\cdots +a_{k}\mathbf {g} _{k},}
8:
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11979:
maps between topological vector spaces are required to be continuous. In particular, the
11867:
11500:
11362:
10878:, as opposed to three space-dimensions—makes it useful for the mathematical treatment of
10074:
10070:
9842:
9832:
8962:
8721:
8227:
7285:
6621:
6556:
6282:
3959:
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2691:
2664:
2643:
1893:
1514:
1450:
623:
552:
542:
393:
293:
285:
276:
259:
255:
243:
138:
19368:
14485:
14093:
14060:
12693:
12499:
11467:
11414:
9728:
9681:
8976:
7729:
5081:; they are then essentially identical as vector spaces, since all identities holding in
4949:
4906:
4117:
3211:
The first example above reduces to this example if an arrow is represented by a pair of
1922:. Equivalently, they are linearly independent if two linear combinations of elements of
19708:
19544:. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press.
19400:
19343:
19319:
19234:
19215:
19130:
19059:
18993:
18430:
17634:
15548:
15532:
15514:
15490:
15412:
14927:
14833:
14682:
14662:
14462:
14442:
14391:
14082:
13996:
12986:{\displaystyle \|\mathbf {x} _{n}\|_{1}=\sum _{i=1}^{2^{n}}2^{-n}=2^{n}\cdot 2^{-n}=1.}
12626:{\displaystyle \mathbf {x} _{n}=\left(2^{-n},2^{-n},\ldots ,2^{-n},0,0,\ldots \right),}
12206:
11873:
11638:
11546:
11228:
11150:
11130:
11110:
11034:
10879:
10080:
10017:
9880:
9815:
9708:
9616:
9471:
8798:
8703:
8683:
8581:
8412:
8389:
8129:
8051:
7862:
7510:
7435:
7379:
7359:
7267:
7133:
7130:. This way, the quotient space "forgets" information that is contained in the subspace
7113:
6780:
6671:
6666:
6649:
6513:
6507:
6489:
6469:
6445:
6425:
6405:
6381:
6357:
6315:
6295:
5776:
5748:
5483:
4929:
4646:
4626:
2612:
2592:
1606:
619:
In this article, vectors are represented in boldface to distinguish them from scalars.
358:
349:
307:
216:
204:
19375:(1971), "Schnelle Multiplikation großer Zahlen (Fast multiplication of big numbers)",
18231:, Carus Mathematical Monographs, Washington, DC: Mathematical Association of America,
15358:
14410:
14020:
11717:
11677:
8578:), where only tuples with finitely many nonzero vectors are allowed. If the index set
6190:
is large enough to contain a zero of this polynomial (which automatically happens for
5164:-component of the arrow, as shown in the image at the right. Conversely, given a pair
19906:
19863:
19790:
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19545:
19512:
19465:
19443:
19392:
19347:
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18278:
18253:
18232:
18214:
18196:
18178:
18160:
18138:
18116:
18098:
18047:
18040:
Differential equations and their applications: an introduction to applied mathematics
18025:
18000:
17966:
17932:
17912:
17894:
17858:
17834:
17826:
17805:
17780:
17754:
17725:
17705:
17667:
17641:
17619:
16123:
16021:
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14153:. Definite values for physical properties such as energy, or momentum, correspond to
14142:
14054:
13921:
13030:
11052:
8763:
6090:
5275:
5149:
5048:
3954:
2581:
2545:
2457:
2129:. It follows that, in general, no base can be explicitly described. For example, the
1961:
under vector addition and scalar multiplication; that is, the sum of two elements of
1578:
1165:
1155:
630:
149:
95:
11495:
of the series depends on the topology imposed on the function space. In such cases,
11354:{\displaystyle \sum _{i=1}^{\infty }f_{i}~=~\lim _{n\to \infty }f_{1}+\cdots +f_{n}}
6332:
that is closed under addition and scalar multiplication (and therefore contains the
6223:
of the map. The set of all eigenvectors corresponding to a particular eigenvalue of
5896:
is the absolute value of the determinant of the 3-by-3 matrix formed by the vectors
3668:
2537:
1525:
19911:
19815:
19668:
19504:
19439:
19435:
19428:
19404:
19384:
19120:
19042:
18985:
18924:
18899:
18847:
18766:
18719:
18586:
18559:
18509:
18441:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
18354:
18013:
17988:
17772:
17697:
17659:
15568:
15457:
15428:
15125:
14945:
14922:
14171:
14167:
14122:
13269:
8724:
which deals with extending notions such as linear maps to several variables. A map
8147:
7154:
2798:
2577:
2572:
on directed line segments that share the same length and direction which he called
2553:
2541:
951:
871:
638:
378:
145:
15883:
It is also common, especially in physics, to denote vectors with an arrow on top:
13875:{\displaystyle \langle f\ ,\ g\rangle =\int _{\Omega }f(x){\overline {g(x)}}\,dx,}
403:
230:
Many vector spaces that are considered in mathematics are also endowed with other
19970:
19763:
19723:
19713:
19555:
19372:
19357:
19290:
19190:
19168:
19092:
19026:
19012:
18957:
18932:
18907:
18888:
18832:
18785:
18657:
18611:
18596:
18528:
18332:
18288:
18078:
18043:
18021:
17996:
17962:
17926:
17890:
17889:, Graduate Texts in Mathematics, vol. 135 (2nd ed.), Berlin, New York:
17872:
17815:
17748:
17719:
15461:
14850:
to obtain an algebra. As a vector space, it is spanned by symbols, called simple
14652:
14395:
14383:
14234:
14126:
11628:
11488:
10612:
10215:{\textstyle |\mathbf {v} |:={\sqrt {\langle \mathbf {v} ,\mathbf {v} \rangle }}.}
9836:
9807:{\displaystyle u(\mathbf {v} \otimes \mathbf {w} )=g(\mathbf {v} ,\mathbf {w} ).}
6506:. Expressed in terms of elements, the span is the subspace consisting of all the
6275:
6256:
3971:
3584:
3570:
3428:, form a vector space over the reals with the usual addition and multiplication:
3385:
2671:
2631:
2588:
2529:
2325:{\displaystyle \mathbf {v} =a_{1}\mathbf {b} _{1}+\cdots +a_{n}\mathbf {b} _{n},}
2134:
2126:
1933:
922:
642:
470:
464:
451:
431:
422:
388:
325:
239:
212:
134:
19109:
Halpern, James D. (Jun 1966), "Bases in Vector Spaces and the Axiom of Choice",
15840:
The set of one-dimensional subspaces of a fixed finite-dimensional vector space
15350:
14944:
varies. The multiplication is given by concatenating such symbols, imposing the
11542:
consist of plane vectors of norm 1. Depicted are the unit spheres in different
6553:-dimensional vector space, any subspace of dimension 1 less, i.e., of dimension
6265:
3573:
provide another class of examples of vector spaces, particularly in algebra and
2548:
by identifying solutions to an equation of two variables with points on a plane
2026:, in the sense that it is the intersection of all linear subspaces that contain
19975:
19896:
19631:
19537:
19411:
19207:
19084:
19004:
18855:
18843:
18818:
18810:
18732:
18607:
18545:(1833), "Sopra alcune applicazioni di un nuovo metodo di geometria analitica",
18472:
18307:
18266:
18245:
18086:
17680:
15925:
15382:
15279:
15146:
14796:
11462:
11060:
10452:
8239:
7049:
6748:{\displaystyle \mathbf {v} +W=\{\mathbf {v} +\mathbf {w} :\mathbf {w} \in W\},}
5893:
4414:
3663:
3425:
2794:
2675:
2659:
2651:
1542:
1119:
1037:
512:
220:
196:
169:
18572:
17776:
17701:
17663:
10087:, which measures angles between vectors. Norms and inner products are denoted
20048:
20008:
19931:
19891:
19858:
19838:
19516:
19461:
19396:
19067:
18781:
18708:
Moore, Gregory H. (1995), "The axiomatization of linear algebra: 1875–1940",
18521:
18488:
18456:
18404:
18389:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
18378:
18359:
18193:
Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets
17615:
17607:
17558:
16293:
15857:
15544:
15510:
15386:
15156:
15142:
14789:
14387:
13746:
13733:
13728:
13716:
12083:
10084:
9850:
5939:
5888:
3563:
2822:
2703:
2683:
2679:
2627:
1151:
844:
809:
398:
363:
320:
263:
19567:
18974:
Eisenberg, Murray; Guy, Robert (1979), "A proof of the hairy ball theorem",
9839:, since the addition operation allows only finitely many terms to be added.
7349:{\displaystyle \operatorname {im} (f)=\{f(\mathbf {v} ):\mathbf {v} \in V\}}
5401:. However, there is no "canonical" or preferred isomorphism; an isomorphism
5303:; the map is an isomorphism if and only if the space is finite-dimensional.
4319:{\displaystyle A={\begin{bmatrix}1&3&1\\4&2&2\end{bmatrix}}}
2973:
A second key example of a vector space is provided by pairs of real numbers
2536:
in the plane or three-dimensional space. Around 1636, French mathematicians
2204:{\displaystyle (\mathbf {b} _{1},\mathbf {b} _{2},\ldots ,\mathbf {b} _{n})}
19830:
19780:
18724:
18591:
18513:
17882:
17128:
16938:
16842:
16332:
15959:
15605:. In particular, a vector space is an affine space over itself, by the map
15578:
15564:
15404:
14174:
acting on functions in terms of these eigenfunctions and their eigenvalues.
14150:
12195:{\displaystyle \mathbf {x} =\left(x_{1},x_{2},\ldots ,x_{n},\ldots \right)}
12078:
12072:
11257:
11059:. Compatible here means that addition and scalar multiplication have to be
8792:
8598:
is finite, the two constructions agree, but in general they are different.
7848:{\displaystyle f\mapsto D(f)=\sum _{i=0}^{n}a_{i}{\frac {d^{i}f}{dx^{i}}},}
5979:
3672:
Addition of functions: the sum of the sine and the exponential function is
3404:
2990:
2068:
572:
337:
267:
130:
31:
19141:
Hughes-Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. (2013),
18564:Éléments d'histoire des mathématiques (Elements of history of mathematics)
18532:
15928:, which is an additional operation on some specific vector spaces, called
13711:
Imposing boundedness conditions not only on the function, but also on its
12865:{\displaystyle \|\mathbf {x} _{n}\|_{\infty }=\sup(2^{-n},0)=2^{-n}\to 0,}
9835:
to another function. Likewise, linear algebra is not adapted to deal with
37:"Linear space" redirects here. For a structure in incidence geometry, see
19921:
19886:
19843:
19688:
16280:
15963:
15524:
15247:
14403:
14188:
12714:
10262:
9931:
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6219:, a basis consisting of eigenvectors. This phenomenon is governed by the
5928:
5452:
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4973:
2840:
2130:
2122:
2101:
1993:
1111:
889:
562:
557:
446:
436:
410:
247:
79:
18213:(2nd ed.), Harlow, Essex, England: Prentice-Hall (published 2002),
17546:
15958:
This is typically the case when a vector space is also considered as an
15523:
shows; those modules that do (including all vector spaces) are known as
14193:
8958:
2766:
2670:
An important development of vector spaces is due to the construction of
19950:
19693:
19388:
19311:
19134:
18997:
18324:
18299:
17793:
16950:
16914:
15440:
14753:
14351:
14162:
14154:
13712:
10608:
7856:
6581:
6422:, when the ambient space is unambiguously a vector space. Subspaces of
6216:
5062:
4703:
2635:
2600:
2533:
2098:
Basis (linear algebra) § Proof that every vector space has a basis
1222:{\displaystyle \mathbf {v} -\mathbf {w} =\mathbf {v} +(-\mathbf {w} ).}
312:
188:
19508:
17909:
Abstract Algebra with Applications: Volume 1: Vector spaces and groups
17294:
17246:
16427:
15547:. The algebro-geometric interpretation of commutative rings via their
11859:{\displaystyle \lim _{n\to \infty }|\mathbf {v} _{n}-\mathbf {v} |=0.}
8122:
The existence of kernels and images is part of the statement that the
7925:, for example). Since differentiation is a linear procedure (that is,
3598:-vector space, by the given multiplication and addition operations of
1074:
Distributivity of scalar multiplication with respect to field addition
1040:
of scalar multiplication with respect to vector addition
19748:
17804:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
16902:
15860:
generalize this by parametrizing linear subspaces of fixed dimension
15830:
15539:, with the elements being called vectors. Some authors use the term
15493:
what vector spaces are to fields: the same axioms, applied to a ring
15151:
14198:
11510:
8532:
8407:
5766:
5344:
5058:
3928:
2647:
2639:
567:
373:
330:
298:
19125:
18989:
17146:
5108:
4457:
is the zero vector. In a similar vein, the solutions of homogeneous
2900:
is defined as the arrow pointing in the opposite direction instead.
104:, can be added together and multiplied ("scaled") by numbers called
19916:
16127:
15447:
15423:
15416:
13222:
12090:
11706:
can be uniformly approximated by a sequence of polynomials, by the
11225:
To make sense of specifying the amount a scalar changes, the field
10875:
5379:
isomorphism) by its dimension, a single number. In particular, any
3950:
2806:
2801:, starting at one fixed point. This is used in physics to describe
2730:
2695:
2616:
368:
208:
157:
44:
19416:
A Comprehensive Introduction to Differential Geometry (Volume Two)
19011:, Graduate Texts in Mathematics, vol. 150, Berlin, New York:
17366:
17342:
16962:
16806:
15573:
14145:
describes the change of physical properties in time by means of a
4326:
is the matrix containing the coefficients of the given equations,
3988:
are closely tied to vector spaces. For example, the solutions of
19600:
17848:
17414:
17330:
17270:
17094:
15287:
11543:
10404:
this reflects the common notion of the angle between two vectors
6109:
is finite-dimensional, this can be rephrased using determinants:
2687:
2655:
83:
17486:
17158:
15811:
in this equation. The space of solutions is the affine subspace
11506:
10064:
8924:{\displaystyle \mathbf {v} \mapsto g(\mathbf {v} ,\mathbf {w} )}
6920:{\displaystyle \left(\mathbf {v} _{1}+\mathbf {v} _{2}\right)+W}
6209:) any linear map has at least one eigenvector. The vector space
1918:
can be written as a linear combination of the other elements of
980:
Compatibility of scalar multiplication with field multiplication
129:
are kinds of vector spaces based on different kinds of scalars:
19926:
17771:, Undergraduate Texts in Mathematics (3rd ed.), Springer,
14851:
12203:
11031:
Convergence questions are treated by considering vector spaces
8997:
6986:{\displaystyle a\cdot (\mathbf {v} +W)=(a\cdot \mathbf {v} )+W}
6290:
3936:
1833:{\displaystyle \mathbf {g} _{1},\ldots ,\mathbf {g} _{k}\in G.}
1130:
are also commonly considered. Such a vector space is called an
302:
17833:(3rd ed.), American Mathematical Soc., pp. 193–222,
17390:
17378:
17194:
16986:
16890:
16866:
10611:. An important variant of the standard dot product is used in
19140:
18627:
Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik
17210:
17058:
17034:
17022:
17010:
16998:
16830:
16595:, ch. "Algèbre linéaire et algèbre multilinéaire", pp. 78–91.
16331:, Corollary 8.3. The sections of the tangent bundle are just
15712:
15527:. Nevertheless, a vector space can be compactly defined as a
9541:
7527:. This concept also extends to linear differential equations
5376:
3379:
3236:
2802:
2549:
2453:
758:
646:
153:
117:
17070:
16854:
16415:
16371:
15176:
is a family of vector spaces parametrized continuously by a
8072:. In particular, the solutions to the differential equation
6161:{\displaystyle \det(f-\lambda \cdot \operatorname {Id} )=0.}
4541:{\displaystyle f^{\prime \prime }(x)+2f^{\prime }(x)+f(x)=0}
3349:
is the above-mentioned simplest example, in which the field
2552:. To achieve geometric solutions without using coordinates,
1973:. This implies that every linear combination of elements of
18195:, Texts in Applied Mathematics, New York: Springer-Verlag,
17426:
17354:
16718:
16020:
This requirement implies that the topology gives rise to a
15966:, while an affine subspace does not necessarily contain it.
15829:
is the space of solutions of the homogeneous equation (the
13029:
are endowed with a norm that replaces the above sum by the
11055:, a structure that allows one to talk about elements being
10600:{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =0}
3604:. For example, the complex numbers are a vector space over
1520:
17402:
17258:
15932:. Scalar multiplication is the multiplication of a vector
15683:
is a vector space, then an affine subspace is a subset of
10152:{\displaystyle \langle \mathbf {v} ,\mathbf {w} \rangle ,}
9821:
8969:
The tensor product is a particular vector space that is a
8219:{\displaystyle V/\ker(f)\;\equiv \;\operatorname {im} (f)}
6533:
Linear subspace of dimension 1 and 2 are referred to as a
5359:
gives rise to a linear map that maps any basis element of
5332:
is expressed uniquely as a linear combination of them. If
2034:
is also the set of all linear combinations of elements of
1231:
Direct consequences of the axioms include that, for every
18573:"A general outline of the genesis of vector space theory"
17498:
17462:
17450:
17106:
17046:
16818:
15036:{\displaystyle \mathbf {v} _{2}\otimes \mathbf {v} _{1}.}
12995:
More generally than sequences of real numbers, functions
11365:
of the corresponding finite partial sums of the sequence
10812:{\displaystyle \langle \mathbf {x} |\mathbf {x} \rangle }
5885:
is uniquely represented by a matrix via this assignment.
5457:
3355:
is also regarded as a vector space over itself. The case
18873:, Contemporary Mathematics volume 31, Providence, R.I.:
18638:, translated by Kannenberg, Lloyd C., Providence, R.I.:
17658:, vol. 242, Springer Science & Business Media,
17522:
17474:
17318:
17282:
17182:
16974:
16878:
16694:
16480:
16478:
14989:{\displaystyle \mathbf {v} _{1}\otimes \mathbf {v} _{2}}
5996:, are particularly important since in this case vectors
3707:{\displaystyle \sin +\exp :\mathbb {R} \to \mathbb {R} }
2854:, but is dilated or shrunk by multiplying its length by
1541:(blue) expressed in terms of different bases: using the
17696:, Applied and Numerical Harmonic Analysis, Birkhäuser,
17438:
16926:
16794:
16782:
16670:
16451:
15585:. It is a two-dimensional subspace shifted by a vector
14830:
is a formal way of adding products to any vector space
11623:
The bigger diamond depicts points of 1-norm equal to 2.
11461:
could be (real or complex) functions belonging to some
9926:
can be ordered by comparing its vectors componentwise.
9877:
under which some vectors can be compared. For example,
8041:{\displaystyle (c\cdot f)^{\prime }=c\cdot f^{\prime }}
7983:{\displaystyle (f+g)^{\prime }=f^{\prime }+g^{\prime }}
5152:
can be expressed as an ordered pair by considering the
2505:
addition and scalar multiplication, whose dimension is
2100:). Moreover, all bases of a vector space have the same
137:. Scalars can also be, more generally, elements of any
17306:
17170:
16622:
16494:
16310:
which restricts to linear isomorphisms between fibers.
10167:
10083:, a datum which measures lengths of vectors, or by an
8965:
depicting the universal property of the tensor product
8541:
8494:
8379:{\displaystyle \left(\mathbf {v} _{i}\right)_{i\in I}}
8269:
7894:
5751:
4708:
The relation of two vector spaces can be expressed by
4478:
4275:
207:. Finite-dimensional vector spaces occur naturally in
18904:
Elements of Mathematics : Algebra I Chapters 1-3
16658:
16240:
16212:
16172:
16145:
16047:
15989:
15924:
Scalar multiplication is not to be confused with the
15889:
15787:
15750:
15611:
15197:
15053:
15002:
14958:
14930:
14859:
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14804:
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13794:
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9903:
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8657:
8621:
8584:
8464:
8435:
8415:
8392:
8342:
8315:
8268:
8172:
8132:
8078:
8054:
7996:
7931:
7885:
7865:
7755:
7732:
7705:
7533:
7513:
7480:
7458:
7438:
7405:
7382:
7362:
7293:
7270:
7248:
7226:
7194:
7162:
7136:
7116:
7087:
7058:
7042:{\displaystyle \mathbf {v} _{1}+W=\mathbf {v} _{2}+W}
6999:
6933:
6873:
6838:
6803:
6783:
6761:
6694:
6674:
6652:
6624:
6598:
6559:
6516:
6492:
6472:
6448:
6428:
6408:
6384:
6360:
6338:
6318:
6298:
6127:
5823:
5799:
5779:
5518:
5486:
4952:
4932:
4909:
4887:
4865:
4726:
4669:
4649:
4629:
4556:
4469:
4423:
4394:
4354:
4332:
4263:
4225:
4179:
4142:
4120:
3994:
3855:
3823:
3720:
3678:
3616:
3256:
3011:
2694:
began to interact, notably with key concepts such as
2634:
which allows for harmonization and simplification of
2472:
2388:
2382:, and that this decomposition is unique. The scalars
2338:
2257:
2229:
2146:
1846:
1787:
1735:
1637:
1479:
1453:
1420:
1371:
1334:
1297:
1263:
1237:
1177:
18469:
Topological vector spaces, distributions and kernels
17570:
17534:
17134:
17082:
16646:
8931:
is linear in the sense above and likewise for fixed
7452:. The kernel of this map is the subspace of vectors
6281:
is a linear subspace. It is the intersection of two
3223:
The simplest example of a vector space over a field
48:
Vector addition and scalar multiplication: a vector
19458:
Lie groups, Lie algebras, and their representations
19306:
18634:Grassmann, Hermann (2000), Kannenberg, L.C. (ed.),
17510:
16598:
16463:
16439:
16111:{\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}}
15781:on linear equations, which can be found by setting
15777:generalizing the homogeneous case discussed in the
15505:. For example, modules need not have bases, as the
15043:Forcing two such elements to be equal leads to the
11674:is not complete because any continuous function on
10777:In contrast to the standard dot product, it is not
7399:An important example is the kernel of a linear map
6688:") is defined as follows: as a set, it consists of
2587:Vectors were reconsidered with the presentation of
2133:form an infinite-dimensional vector space over the
19427:
19233:
18789:
18271:Introductory functional analysis with applications
17633:
16706:
16634:
16610:
16383:
16359:
16262:
16227:
16185:
16158:
16110:
16003:
15907:
15803:
15769:
15671:
15215:
15116:
15035:
14988:
14936:
14913:
14842:
14822:
14780:
14744:
14691:
14671:
14643:
14527:
14471:
14451:
14431:
14374:
14320:
14295:
14225:
14105:
14072:
14041:
14005:
13978:
13948:
13912:
13874:
13780:
13701:
13583:
13545:
13516:
13382:
13333:
13258:
13213:
13171:
13142:
13021:
12985:
12864:
12767:
12738:
12705:
12682:
12652:
12625:
12511:
12488:
12459:
12345:
12254:
12215:
12194:
12110:
12050:
12028:
12000:
11971:
11934:
11902:
11882:
11858:
11789:
11767:
11738:
11698:
11662:
11615:
11592:
11555:
11534:
11479:
11453:
11426:
11403:
11353:
11237:
11217:
11189:
11159:
11139:
11119:
11099:
11077:
11043:
11010:
10866:
10811:
10769:
10636:
10599:
10557:
10443:
10418:
10396:
10364:
10253:
10223:Vector spaces endowed with such data are known as
10214:
10151:
10111:
10053:
10026:
10006:
9979:
9918:
9889:
9869:
9806:
9740:
9717:
9693:
9670:
9625:
9601:
9571:
9532:
9506:
9480:
9460:
9104:
8988:
8948:
8923:
8876:
8854:
8829:
8807:
8783:
8754:
8712:
8692:
8672:
8643:
8590:
8570:
8523:
8477:
8450:
8421:
8398:
8378:
8328:
8301:
8218:
8138:
8099:
8060:
8040:
7982:
7917:
7871:
7847:
7741:
7718:
7691:
7519:
7499:
7466:
7444:
7424:
7388:
7368:
7348:
7276:
7256:
7234:
7212:
7180:
7142:
7122:
7102:
7073:
7041:
6985:
6919:
6859:
6824:
6789:
6769:
6747:
6680:
6658:
6638:
6610:
6571:
6522:
6498:
6478:
6454:
6434:
6414:
6390:
6366:
6346:
6324:
6304:
6160:
5846:{\displaystyle \mathbf {x} \mapsto A\mathbf {x} .}
5845:
5807:
5785:
5757:
5737:
5492:
4961:
4938:
4918:
4895:
4873:
4851:
4682:
4655:
4635:
4615:
4540:
4449:
4405:
4381:
4340:
4318:
4247:
4205:
4165:
4129:
4106:
3915:
3841:
3783:
3706:
3640:
3307:
3201:
2959:has the opposite direction and the same length as
2485:
2420:
2370:
2324:
2243:
2203:
1878:
1832:
1773:
1721:
1498:
1465:
1439:
1405:
1356:
1319:
1280:
1249:
1221:
614:
168:. The concept of vector spaces is fundamental for
19367:
18384:
18252:(6th ed.), New York: John Wiley & Sons,
18209:Ifeachor, Emmanuel C.; Jervis, Barrie W. (2001),
16526:
14396:rings of functions of algebraic geometric objects
14335:defining the multiplication of two vectors is an
13566:
13334:{\displaystyle f_{1},f_{2},\ldots ,f_{n},\ldots }
13238:
12469:The topologies on the infinite-dimensional space
12119:consisting of infinite vectors with real entries
8302:{\displaystyle \textstyle {\prod _{i\in I}V_{i}}}
7425:{\displaystyle \mathbf {x} \mapsto A\mathbf {x} }
3945:. Many notions in topology and analysis, such as
2989:is significant, so such a pair is also called an
2965:(blue vector pointing down in the second image).
20046:
19112:Proceedings of the American Mathematical Society
18864:"Existence of bases implies the axiom of choice"
18547:Il poligrafo giornale di scienze, lettre ed arti
18092:
17336:
13599:
13398:
12809:
12291:
11805:
11307:
9701:shown in the diagram with a dotted arrow, whose
9609:is bilinear. The universality states that given
9602:{\displaystyle \mathbf {v} \otimes \mathbf {w} }
8233:
6250:
6128:
3784:{\displaystyle (\sin +\exp )(x)=\sin(x)+\exp(x)}
2793:The first example of a vector space consists of
64:is stretched by a factor of 2, yielding the sum
18842:
18211:Digital Signal Processing: A Practical Approach
18190:
17953:
17600:Elementary Linear Algebra: Applications Version
16682:
16421:
16347:
15962:. In this case, a linear subspace contains the
15558:
15555:, the algebraic counterpart to vector bundles.
8158:. Because of this, many statements such as the
5968:
2223:. The definition of a basis implies that every
1406:{\displaystyle (-1)\mathbf {v} =-\mathbf {v} ,}
649:listed below. In this context, the elements of
19039:Classic Set Theory: A guided independent study
18780:
18385:Narici, Lawrence; Beckenstein, Edward (2011).
18208:
17961:(in German) (9th ed.), Berlin, New York:
15825:is a particular solution of the equation, and
5219:between two vector spaces form a vector space
19616:
19480:"The JPEG still picture compression standard"
18429:
18351:An introduction to abstract harmonic analysis
17995:, Elements of mathematics, Berlin, New York:
17216:
17152:
16319:A line bundle, such as the tangent bundle of
15359:identifying open intervals with the real line
15353:can be seen as a line bundle over the circle
14952:. In general, there are no relations between
10819:also takes negative values, for example, for
10065:Normed vector spaces and inner product spaces
1171:Subtraction of two vectors can be defined as
595:
219:infinite-dimensional vector spaces, and many
18973:
18616:(in French), Chez Firmin Didot, père et fils
18093:Dennery, Philippe; Krzywicki, Andre (1996),
17504:
16099:
16092:
16080:
16073:
16061:
16048:
15415:, there is no (tangent) vector field on the
14161:and the associated wavefunctions are called
13813:
13795:
13362:
13348:
13268:These spaces are complete. (If one uses the
13193:
13186:
13046:
13039:
12895:
12879:
12797:
12781:
12368:
12359:
12278:
12269:
11020:
10907:
10889:
10806:
10788:
10669:
10651:
10588:
10572:
10288:
10272:
10204:
10188:
10143:
10127:
10079:"Measuring" vectors is done by specifying a
9840:
7343:
7312:
6739:
6709:
5965:if and only if its determinant is positive.
5299:, any vector space can be embedded into its
5008:, which is a map such that the two possible
3332:form a vector space that is usually denoted
757:To have a vector space, the eight following
19455:
18191:Gasquet, Claude; Witomski, Patrick (1999),
17597:
17444:
16516:
15551:allows the development of concepts such as
14237:of functions on this hyperbola is given by
13741:Complete inner product spaces are known as
11631:has a limit; such a vector space is called
9572:{\displaystyle (\mathbf {v} ,\mathbf {w} )}
6993:. The key point in this definition is that
4697:
4248:{\displaystyle A\mathbf {x} =\mathbf {0} ,}
2968:
2925:, but is stretched to the double length of
2848:, the arrow that has the same direction as
2438:on the basis. They are also said to be the
1357:{\displaystyle s\mathbf {0} =\mathbf {0} ,}
1320:{\displaystyle 0\mathbf {v} =\mathbf {0} ,}
19623:
19609:
19257:
19187:Riemannian Geometry and Geometric Analysis
19162:
18749:: CS1 maint: location missing publisher (
18541:
17850:Matrix Analysis and Applied Linear Algebra
16628:
16328:
15848:; it may be used to formalize the idea of
15770:{\displaystyle A\mathbf {v} =\mathbf {b} }
15723:) and consists of all vectors of the form
15687:obtained by translating a linear subspace
15630:
14303:an infinite-dimensional vector space over
14296:{\displaystyle \mathbf {R} /(x\cdot y-1),}
13565:
13237:
9845:require considering additional structures.
8200:
8196:
7500:{\displaystyle A\mathbf {x} =\mathbf {0} }
5415:is equivalent to the choice of a basis of
5065:). If there exists an isomorphism between
3641:{\displaystyle \mathbf {Q} (i{\sqrt {5}})}
3393:, numbers that can be written in the form
3380:Complex numbers and other field extensions
3308:{\displaystyle (a_{1},a_{2},\dots ,a_{n})}
2565:
1499:{\displaystyle \mathbf {v} =\mathbf {0} .}
1440:{\displaystyle s\mathbf {v} =\mathbf {0} }
602:
588:
114:must satisfy certain requirements, called
19498:
19487:IEEE Transactions on Consumer Electronics
19124:
19058:
18817:, Advanced Book Classics (2nd ed.),
18723:
18633:
18620:
18590:
18358:
17944:
17870:
17687:, vol. 7, Princeton University Press
17396:
17076:
17028:
16860:
16664:
16484:
15804:{\displaystyle \mathbf {b} =\mathbf {0} }
14166:
13862:
13675:
13490:
13383:{\displaystyle \|f_{n}\|_{p}<\infty ,}
13103:
13022:{\displaystyle f:\Omega \to \mathbb {R} }
13015:
11190:{\displaystyle \mathbf {x} +\mathbf {y} }
9866:
9859:
7918:{\displaystyle f^{\prime \prime }(x)^{2}}
4093:
4082:
4076:
4071:
4064:
4057:
4037:
4029:
4023:
4018:
4011:
4004:
3700:
3692:
2137:, for which no specific basis is known.
1989:of linear subspaces is a linear subspace.
1509:Even more concisely, a vector space is a
1007:Identity element of scalar multiplication
183:Vector spaces are characterized by their
19286:Categories for the Working Mathematician
19279:
19231:
19206:
19143:Calculus : Single and Multivariable
19003:
18923:
18898:
18675:
18558:
18480:
18265:
18244:
18012:
17987:
17924:
17906:
17825:
17746:
17492:
17432:
17384:
17240:
17228:
17204:
17164:
17088:
16756:
16740:
16652:
16592:
16433:
16401:
16025:
15572:
15150:
14192:
13732:
11505:
10565:Because of this, two vectors satisfying
8957:
6927:, and scalar multiplication is given by
6264:
5887:
5456:
5107:
3667:
1774:{\displaystyle a_{1},\ldots ,a_{k}\in F}
1524:
1521:Bases, vector coordinates, and subspaces
110:. The operations of vector addition and
43:
19477:
19422:
19108:
19036:
18948:
18606:
18527:
18172:
18077:
17694:A Basis Theory Primer: Expanded Edition
17653:
17576:
17540:
17372:
17252:
16604:
15183:. More precisely, a vector bundle over
14178:
13221:and equipped with this norm are called
10867:{\displaystyle \mathbf {x} =(0,0,0,1).}
9822:Vector spaces with additional structure
6002:can be compared with their image under
5130:yields an isomorphism of vector spaces.
2903:The following shows a few examples: if
1983:The closure property also implies that
14:
20047:
20014:Comparison of linear algebra libraries
19542:An introduction to homological algebra
19536:
19410:
19083:
18809:
18689:
18570:
18487:
18463:
18348:
18226:
18128:
18110:
17679:
17602:(10th ed.), John Wiley & Sons
17516:
17480:
17324:
17300:
17288:
17264:
17200:
16896:
16724:
16712:
16640:
16616:
16576:
16194:
15361:). It is, however, different from the
15321:) is isomorphic to the trivial bundle
8068:) this assignment is linear, called a
6242:
5365:to the corresponding basis element of
4616:{\displaystyle f(x)=ae^{-x}+bxe^{-x},}
2720:
2557:
1623:, a linear combination of elements of
19604:
19337:
18861:
18773:
18731:
18707:
18411:
18175:Fourier Analysis and Its Applications
18150:
18059:
18037:
17881:
17846:
17737:
17721:Vector Spaces and Matrices in Physics
17631:
17606:
17598:Anton, Howard; Rorres, Chris (2010),
17564:
17552:
17528:
17312:
17188:
17176:
17140:
17124:
17100:
17064:
17040:
17016:
17004:
16992:
16968:
16920:
16836:
16728:
16700:
16676:
16544:
16532:
16500:
16469:
16457:
16445:
16409:
16389:
16377:
16365:
16323:is trivial if and only if there is a
15976:
15131:
14823:{\displaystyle \operatorname {T} (V)}
14659:Examples include the vector space of
13214:{\displaystyle \|f\|_{p}<\infty ,}
13179:(for example an interval) satisfying
12519:For example, the sequence of vectors
12255:{\displaystyle (1\leq p\leq \infty )}
12086:, are complete normed vector spaces.
11167:vary by a bounded amount, then so do
8524:{\textstyle \bigoplus _{i\in I}V_{i}}
6466:, and it is the smallest subspace of
6233:corresponding to the eigenvalue (and
2772:Scalar multiplication: the multiples
2448:on the basis. One also says that the
27:Algebraic structure in linear algebra
19260:Optimization by vector space methods
19184:
18323:
18298:
18273:, Wiley Classics Library, New York:
17792:
17766:
17717:
17691:
17468:
17456:
17420:
17408:
17360:
17348:
17276:
17112:
17052:
16980:
16956:
16944:
16932:
16908:
16884:
16872:
16848:
16824:
16812:
16800:
16788:
16772:
16560:
16512:
16353:
15317:such that the restriction of π to π(
4114:are given by triples with arbitrary
2603:by the latter. They are elements in
1577:(black), and using a different, non-
19340:The geometry of Minkowski spacetime
18852:Introduction to Commutative Algebra
18655:
17740:Foundations of Discrete Mathematics
16688:
14949:
13259:{\displaystyle L^{\;\!p}(\Omega ).}
12008:consists of continuous functionals
11404:{\displaystyle f_{1},f_{2},\ldots }
10874:Singling out the fourth coordinate—
8571:{\textstyle \coprod _{i\in I}V_{i}}
5421:, by mapping the standard basis of
3965:
3916:{\displaystyle (f+g)(w)=f(w)+g(w),}
3376:) reduces to the previous example.
3218:
2421:{\displaystyle a_{1},\ldots ,a_{n}}
2110:Dimension theorem for vector spaces
2018:is the smallest linear subspace of
1879:{\displaystyle a_{1},\ldots ,a_{k}}
1118:, and when the scalar field is the
24:
19630:
19289:(2nd ed.), Berlin, New York:
19189:(4th ed.), Berlin, New York:
19167:(3rd ed.), Berlin, New York:
18956:(2nd ed.), Berlin, New York:
17945:Stoll, R. R.; Wong, E. T. (1968),
16254:
15403:, since there is a global nonzero
15239:) is a vector space. The case dim
14805:
13990:, established an approximation of
13913:{\displaystyle {\overline {g(x)}}}
13824:
13769:
13619:
13609:
13584:{\displaystyle L^{\;\!p}(\Omega )}
13575:
13427:
13417:
13374:
13247:
13205:
13166:
13069:
13008:
12801:
12730:
12451:
12332:
12282:
12246:
11897:
11815:
11607:
11317:
11282:
10489:
10261:can be equipped with the standard
8336:consists of the set of all tuples
8033:
8014:
7975:
7962:
7949:
7891:
6860:{\displaystyle \mathbf {v} _{2}+W}
6825:{\displaystyle \mathbf {v} _{1}+W}
6229:forms a vector space known as the
5855:Moreover, after choosing bases of
5200:is negative) turns back the arrow
4503:
4475:
4450:{\displaystyle \mathbf {0} =(0,0)}
3657:
2931:(the second image). Equivalently,
2829:of the two arrows, and is denoted
2371:{\displaystyle a_{1},\dots ,a_{n}}
1281:{\displaystyle \mathbf {v} \in V,}
60:(red, upper illustration). Below,
54:(blue) is added to another vector
25:
20086:
20070:Vectors (mathematics and physics)
19576:
19064:Introduction to Quantum Mechanics
18977:The American Mathematical Monthly
15983:and derive the concrete shape of
15136:
14781:{\displaystyle \mathbf {R} ^{3},}
13722:
12029:{\displaystyle V\to \mathbf {R} }
11935:{\displaystyle \mathbf {R} ^{2}:}
11708:Weierstrass approximation theorem
10644:endowed with the Lorentz product
10397:{\displaystyle \mathbf {R} ^{2},}
9671:{\displaystyle g:V\times W\to X,}
8720:is one of the central notions of
8601:
6588:The counterpart to subspaces are
5182:to the right (or to the left, if
5138:) are isomorphic: a planar arrow
5093:, transported to similar ones in
4461:form vector spaces. For example,
3550:as representing the ordered pair
2244:{\displaystyle \mathbf {v} \in V}
1965:and the product of an element of
254:, which include function spaces,
20027:
20026:
20004:Basic Linear Algebra Subprograms
19762:
19418:, Houston, TX: Publish or Perish
19236:Advanced Engineering Mathematics
19089:Finite-dimensional vector spaces
18613:Théorie analytique de la chaleur
18250:Advanced Engineering Mathematics
17955:van der Waerden, Bartel Leendert
17874:Linear Algebra with Applications
17750:Advanced Engineering Mathematics
17685:Finite Dimensional Vector Spaces
17654:Grillet, Pierre Antoine (2007),
16313:
16286:
15852:lines intersecting at infinity.
15797:
15789:
15763:
15755:
15662:
15654:
15643:
15635:
15191:equipped with a continuous map
15104:
15089:
15071:
15056:
15020:
15005:
14976:
14961:
14898:
14877:
14862:
14765:
14311:
14245:
14121:, it enables one to construct a
14113:its cardinality is known as the
12884:
12786:
12528:
12363:
12273:
12127:
12066:
12044:
12022:
11943:spaces without additional data.
11919:
11841:
11827:
11783:
11768:{\displaystyle \mathbf {v} _{n}}
11755:
11535:{\displaystyle \mathbf {R} ^{2}}
11522:
11208:
11183:
11175:
11093:
11071:
10903:
10893:
10827:
10802:
10792:
10665:
10655:
10637:{\displaystyle \mathbf {R} ^{4}}
10624:
10584:
10576:
10543:
10525:
10504:
10496:
10471:
10463:
10434:
10412:
10381:
10303:
10295:
10284:
10276:
10200:
10192:
10174:
10139:
10131:
10100:
9919:{\displaystyle \mathbf {R} ^{n}}
9906:
9794:
9786:
9769:
9761:
9595:
9587:
9562:
9554:
9468:These rules ensure that the map
9439:
9430:
9416:
9407:
9380:
9365:
9353:
9342:
9328:
9319:
9305:
9286:
9269:
9254:
9213:
9196:
9182:
9171:
9141:
9133:
9089:
9074:
9053:
9038:
9023:
9008:
8939:
8914:
8906:
8892:
8870:
8845:
8823:
8755:{\displaystyle g:V\times W\to X}
8451:{\displaystyle \mathbf {v} _{i}}
8438:
8350:
7493:
7485:
7460:
7418:
7407:
7333:
7322:
7250:
7228:
7103:{\displaystyle \mathbf {v} _{2}}
7090:
7074:{\displaystyle \mathbf {v} _{1}}
7061:
7023:
7002:
6970:
6944:
6896:
6881:
6841:
6806:
6763:
6729:
6721:
6713:
6696:
6340:
5836:
5825:
5801:
5520:
5500:gives rise to a linear map from
5261:. The space of linear maps from
5077:, the two spaces are said to be
4889:
4867:
4838:
4811:
4784:
4767:
4746:
4738:
4425:
4399:
4334:
4238:
4230:
3618:
2765:
2729:
2309:
2278:
2259:
2231:
2188:
2167:
2152:
2092:A subset of a vector space is a
1811:
1790:
1706:
1675:
1650:
1489:
1481:
1433:
1425:
1396:
1385:
1347:
1339:
1310:
1302:
1265:
1209:
1195:
1187:
1179:
677:assigns to any two vectors
19902:Seven-dimensional cross product
18950:Coxeter, Harold Scott MacDonald
18229:A Panorama of Harmonic Analysis
17234:
17222:
17118:
16273:
16200:
16133:
16031:
16014:
15969:
15952:
14017:, every continuous function on
13781:{\displaystyle L^{2}(\Omega ),}
12496:are inequivalent for different
12439:
12320:
11672:topology of uniform convergence
8644:{\displaystyle V\otimes _{F}W,}
8608:Tensor product of vector spaces
8386:, which specify for each index
7879:appear linearly (as opposed to
6797:. The sum of two such elements
2678:. This was later formalized by
1122:, the vector space is called a
1114:, the vector space is called a
615:Definition and basic properties
172:, together with the concept of
18929:General Topology. Chapters 1-4
18153:Partial differential equations
18097:, Courier Dover Publications,
17567:, Exercise 5.13.15–17, p. 442.
16947:, ch. V.3., Corollary, p. 106.
16851:, ch. IV.4, Corollary, p. 106.
16263:{\displaystyle L^{p}(\Omega )}
16257:
16251:
16126:to get a norm, and not only a
15939:
15918:
15896:
15877:
15778:
15650:
15647:
15631:
15621:
15385:consists of the collection of
15294:and some (fixed) vector space
15207:
14817:
14811:
14718:
14706:
14632:
14629:
14617:
14608:
14602:
14599:
14587:
14578:
14572:
14569:
14557:
14548:
14522:
14510:
14501:
14489:
14426:
14414:
14382:forms an algebra known as the
14369:
14363:
14287:
14269:
14261:
14249:
14036:
14024:
13940:
13934:
13901:
13895:
13853:
13847:
13838:
13832:
13772:
13766:
13689:
13683:
13661:
13655:
13639:
13633:
13606:
13578:
13572:
13553:belonging to the vector space
13540:
13534:
13504:
13498:
13476:
13470:
13454:
13448:
13414:
13250:
13244:
13117:
13111:
13093:
13088:
13082:
13075:
13011:
12853:
12834:
12812:
12413:
12397:
12316:
12301:
12249:
12231:
12018:
11960:
11846:
11821:
11812:
11733:
11721:
11693:
11681:
11654:
11642:
11487:in which case the series is a
11314:
11218:{\displaystyle a\mathbf {x} .}
10898:
10858:
10834:
10797:
10660:
10548:
10538:
10530:
10520:
10508:
10492:
10179:
10169:
10112:{\displaystyle |\mathbf {v} |}
10105:
10095:
9980:{\displaystyle f=f^{+}-f^{-}.}
9849:A vector space may be given a
9798:
9782:
9773:
9757:
9659:
9566:
9550:
9390:
9360:
9279:
9249:
9217:
9203:
9175:
9161:
9145:
9129:
8918:
8902:
8896:
8746:
8213:
8207:
8193:
8187:
8088:
8082:
8010:
7997:
7945:
7932:
7906:
7899:
7771:
7765:
7759:
7749:too. In the corresponding map
7411:
7326:
7318:
7306:
7300:
7204:
7175:
7169:
6974:
6960:
6954:
6940:
6149:
6131:
5829:
5575:
5572:
5527:
4842:
4834:
4815:
4801:
4788:
4780:
4771:
4763:
4750:
4734:
4566:
4560:
4529:
4523:
4514:
4508:
4489:
4483:
4444:
4432:
4373:
4355:
3907:
3901:
3892:
3886:
3877:
3871:
3868:
3856:
3836:
3824:
3778:
3772:
3760:
3754:
3742:
3736:
3733:
3721:
3696:
3635:
3622:
3302:
3257:
3189:
3171:
3161:
3149:
3136:
3084:
3074:
3048:
3042:
3016:
2981:. The order of the components
2198:
2147:
1381:
1372:
1213:
1202:
13:
1:
18875:American Mathematical Society
18796:, Toronto: Thomson Learning,
18640:American Mathematical Society
18608:Fourier, Jean Baptiste Joseph
18157:American Mathematical Society
18113:Real analysis and probability
17802:Graduate Texts in Mathematics
17585:
15422:which is everywhere nonzero.
14321:{\displaystyle \mathbf {R} .}
14149:, whose solutions are called
14147:partial differential equation
11710:. In contrast, the space of
10444:{\displaystyle \mathbf {y} ,}
10014:denotes the positive part of
8949:{\displaystyle \mathbf {v} .}
8855:{\displaystyle \mathbf {w} .}
8309:of a family of vector spaces
8234:Direct product and direct sum
8154:) that behaves much like the
6251:Subspaces and quotient spaces
5817:
5135:
4663:are arbitrary constants, and
4463:
4459:linear differential equations
4406:{\displaystyle A\mathbf {x} }
4219:
3795:Functions from any fixed set
2809:. Given any two such arrows,
2686:, around 1920. At that time,
1110:When the scalar field is the
722:, assigns to any scalar
697:which is commonly written as
669:The binary operation, called
98:whose elements, often called
19744:Eigenvalues and eigenvectors
19438:Mathematics Series, London:
18433:; Wolff, Manfred P. (1999).
18329:Real and functional analysis
17907:Spindler, Karlheinz (1993),
17871:Nicholson, W. Keith (2018),
17337:Dennery & Krzywicki 1996
16341:
15979:, choose to start with this
15868:of subspaces, respectively.
15559:Affine and projective spaces
15468:of that bundle are known as
15155:A Möbius strip. Locally, it
14401:Another crucial example are
14350:For example, the set of all
13905:
13857:
13788:with inner product given by
12051:{\displaystyle \mathbf {C} }
11790:{\displaystyle \mathbf {v} }
11503:are two prominent examples.
11100:{\displaystyle \mathbf {y} }
11078:{\displaystyle \mathbf {x} }
10419:{\displaystyle \mathbf {x} }
8877:{\displaystyle \mathbf {w} }
8830:{\displaystyle \mathbf {v} }
8815:is linear in both variables
8070:linear differential operator
7467:{\displaystyle \mathbf {x} }
7257:{\displaystyle \mathbf {0} }
7235:{\displaystyle \mathbf {v} }
6770:{\displaystyle \mathbf {v} }
6611:{\displaystyle W\subseteq V}
6347:{\displaystyle \mathbf {0} }
5975:Eigenvalues and eigenvectors
5969:Eigenvalues and eigenvectors
5808:{\displaystyle \mathbf {x} }
4896:{\displaystyle \mathbf {w} }
4874:{\displaystyle \mathbf {v} }
4692:natural exponential function
4341:{\displaystyle \mathbf {x} }
3986:homogeneous linear equations
3976:Linear differential equation
2993:. Such a pair is written as
2466:on the basis, since the set
761:must be satisfied for every
716:The binary function, called
225:cardinality of the continuum
7:
19589:Encyclopedia of Mathematics
19456:Varadarajan, V. S. (1974),
18699:(in German), archived from
18566:(in French), Paris: Hermann
18416:(2 ed.), McGraw-Hill,
18173:Folland, Gerald B. (1992),
18151:Evans, Lawrence C. (1998),
18111:Dudley, Richard M. (1989),
18067:Encyclopedia of Mathematics
17980:
16422:Atiyah & Macdonald 1969
16327:that vanishes nowhere, see
15908:{\displaystyle {\vec {v}}.}
15701:; this space is denoted by
15378:whereas the former is not.
15216:{\displaystyle \pi :E\to X}
14226:{\displaystyle x\cdot y=1.}
14123:basis of orthogonal vectors
12690:and the following ones are
11980:
8973:recipient of bilinear maps
8673:{\displaystyle V\otimes W,}
6269:A line passing through the
5793:with the coordinate vector
5442:
5112:Describing an arrow vector
3980:Systems of linear equations
2709:
2621:systems of linear equations
2116:
1926:define the same element of
797:
657:, and the elements of
250:. This is also the case of
178:systems of linear equations
10:
20091:
19478:Wallace, G.K. (Feb 1992),
19338:Naber, Gregory L. (2003),
19258:Luenberger, David (1997),
19240:(8th ed.), New York:
19066:, Upper Saddle River, NJ:
18854:, Advanced Book Classics,
18656:Guo, Hongyu (2021-06-16),
18227:Krantz, Steven G. (1999),
18135:Princeton University Press
18095:Mathematics for Physicists
17692:Heil, Christopher (2011),
17636:Matrices and vector spaces
17632:Brown, William A. (1991),
17590:
16959:, Theorem VII.9.8, p. 198.
15562:
15479:
15475:
15393:is globally isomorphic to
15140:
14182:
13726:
12739:{\displaystyle p=\infty ,}
12070:
11024:
10068:
9870:{\displaystyle \,\leq ,\,}
9678:there exists a unique map
9533:{\displaystyle V\otimes W}
8862:That is to say, for fixed
8605:
8237:
8156:category of abelian groups
8107:form a vector space (over
6777:is an arbitrary vector in
6254:
6215:may or may not possess an
6049:is a scalar, is called an
5972:
5446:
4991:such that there exists an
4701:
3969:
3661:
3610:, and the field extension
2919:has the same direction as
2713:
2619:in 1867, who also defined
2532:, via the introduction of
2523:
2456:of the coordinates is the
1890:of the linear combination.
938:, there exists an element
203:, and its dimension is an
148:, which allow modeling of
36:
29:
20022:
19984:
19940:
19877:
19829:
19771:
19760:
19656:
19638:
19163:Husemoller, Dale (1994),
18659:What Are Tensors Exactly?
18571:Dorier, Jean-Luc (1995),
18435:Topological Vector Spaces
18387:Topological Vector Spaces
18349:Loomis, Lynn H. (2011) ,
17993:Topological vector spaces
17777:10.1007/978-1-4757-1949-9
17753:, John Wiley & Sons,
17702:10.1007/978-0-8176-4687-5
17664:10.1007/978-0-387-71568-1
17555:, Example 5.13.5, p. 436.
17217:Schaefer & Wolff 1999
17153:Schaefer & Wolff 1999
16923:, Th. 2.5 and 2.6, p. 49.
14132:The solutions to various
14015:Stone–Weierstrass theorem
13390:satisfying the condition
12489:{\displaystyle \ell ^{p}}
12111:{\displaystyle \ell ^{p}}
11981:(topological) dual space
11250:topological vector spaces
11021:Topological vector spaces
9507:{\displaystyle V\times W}
8784:{\displaystyle V\times W}
8252:of vector spaces and the
8166:in matrix-related terms)
8160:first isomorphism theorem
8124:category of vector spaces
6462:of vectors is called its
6176:characteristic polynomial
2736:Vector addition: the sum
2716:Examples of vector spaces
2560:introduced the notion of
2514:one-to-one correspondence
2108:of the vector space (see
1953:is a non-empty subset of
1898:The elements of a subset
252:topological vector spaces
144:Vector spaces generalize
19232:Kreyszig, Erwin (1999),
19041:(1st ed.), London:
18691:Möbius, August Ferdinand
18129:Dunham, William (2005),
17747:Kreyszig, Erwin (2020),
17505:Eisenberg & Guy 1979
16911:, Theorem IV.2.1, p. 95.
16228:{\displaystyle p\neq 2,}
15871:
15374:, because the latter is
14201:, given by the equation
13992:differentiable functions
13524:there exists a function
11714:continuous functions on
11616:{\displaystyle \infty .}
11027:Topological vector space
9897:-dimensional real space
9841:Therefore, the needs of
7213:{\displaystyle f:V\to W}
5953:corresponding to a real
4698:Linear maps and matrices
4382:{\displaystyle (a,b,c),}
4206:{\displaystyle c=-5a/2.}
2969:Ordered pairs of numbers
2658:. Italian mathematician
2638:. Around the same time,
2528:Vector spaces stem from
2518:vector space isomorphism
2442:of the decomposition of
877:There exists an element
744:, which is denoted
30:Not to be confused with
20065:Mathematical structures
19316:Wheeler, John Archibald
19037:Goldrei, Derek (1996),
18862:Blass, Andreas (1984),
18844:Atiyah, Michael Francis
18811:Atiyah, Michael Francis
18682:Lectures on Quaternions
18677:Hamilton, William Rowan
18501:Fundamenta Mathematicae
17928:Linear Algebraic Groups
17925:Springer, T.A. (2000),
17887:Advanced Linear Algebra
17847:Meyer, Carl D. (2000),
17742:, John Wiley & Sons
17640:, New York: M. Dekker,
17303:, Theorem 11.2, p. 102.
16517:Anton & Rorres 2010
16270:is not a Hilbert space.
15543:to mean modules over a
15503:multiplicative inverses
15280:"trivial" vector bundle
15250:. For any vector space
15187:is a topological space
14745:{\displaystyle =xy-yx,}
14644:{\displaystyle ]+]+]=0}
14439:denotes the product of
14115:Hilbert space dimension
14051:trigonometric functions
14013:by polynomials. By the
13172:{\displaystyle \Omega }
11972:{\displaystyle V\to W,}
11903:{\displaystyle \infty }
7699:where the coefficients
7181:{\displaystyle \ker(f)}
5347:between fixed bases of
5312:is chosen, linear maps
3648:is a vector space over
3575:algebraic number theory
2910:, the resulting vector
2746:(black) of the vectors
2562:barycentric coordinates
2497:-tuples of elements of
1026:multiplicative identity
645:that satisfy the eight
160:, that have not only a
39:Linear space (geometry)
19729:Row and column vectors
18759:Formulario mathematico
18725:10.1006/hmat.1995.1025
18630:(in German), O. Wigand
18592:10.1006/hmat.1995.1024
18514:10.4064/fm-3-1-133-181
18412:Rudin, Walter (1991),
18038:Braun, Martin (1993),
17255:, Proposition III.7.2.
16264:
16229:
16187:
16160:
16112:
16005:
15975:Some authors, such as
15909:
15805:
15771:
15673:
15590:
15470:differential one-forms
15411:. In contrast, by the
15282:. Vector bundles over
15217:
15169:
15118:
15037:
14990:
14938:
14915:
14844:
14824:
14782:
14746:
14693:
14673:
14645:
14529:
14473:
14453:
14433:
14376:
14343:-algebra if the field
14328:
14322:
14297:
14227:
14157:of a certain (linear)
14134:differential equations
14107:
14080:in the sense that the
14074:
14043:
14007:
13980:
13950:
13914:
13876:
13782:
13738:
13703:
13585:
13547:
13518:
13384:
13335:
13272:instead, the space is
13260:
13215:
13173:
13144:
13023:
12987:
12934:
12866:
12769:
12740:
12707:
12684:
12683:{\displaystyle 2^{-n}}
12654:
12627:
12513:
12490:
12461:
12347:
12256:
12217:
12196:
12112:
12052:
12030:
12002:
11973:
11936:
11904:
11884:
11860:
11791:
11769:
11740:
11700:
11664:
11624:
11617:
11594:
11593:{\displaystyle p=1,2,}
11557:
11536:
11481:
11455:
11428:
11405:
11355:
11286:
11239:
11219:
11191:
11161:
11141:
11121:
11101:
11079:
11051:carrying a compatible
11045:
11012:
10868:
10813:
10771:
10638:
10601:
10559:
10445:
10420:
10398:
10366:
10255:
10216:
10153:
10113:
10055:
10028:
10008:
9981:
9920:
9891:
9871:
9808:
9742:
9719:
9695:
9672:
9627:
9603:
9573:
9534:
9508:
9482:
9462:
9106:
8990:
8966:
8950:
8925:
8878:
8856:
8831:
8809:
8785:
8756:
8714:
8694:
8674:
8645:
8592:
8572:
8525:
8479:
8452:
8423:
8400:
8380:
8330:
8303:
8220:
8140:
8101:
8100:{\displaystyle D(f)=0}
8062:
8042:
7984:
7919:
7873:
7849:
7797:
7743:
7720:
7693:
7521:
7501:
7468:
7446:
7432:for some fixed matrix
7426:
7390:
7370:
7350:
7278:
7258:
7236:
7214:
7182:
7144:
7124:
7104:
7075:
7043:
6987:
6921:
6861:
6826:
6791:
6771:
6749:
6682:
6660:
6640:
6612:
6590:quotient vector spaces
6573:
6524:
6500:
6480:
6456:
6436:
6416:
6392:
6368:
6348:
6326:
6306:
6286:
6162:
5963:orientation preserving
5924:
5847:
5809:
5787:
5759:
5739:
5703:
5650:
5603:
5494:
5462:
5131:
4963:
4940:
4920:
4897:
4875:
4853:
4684:
4657:
4637:
4617:
4542:
4451:
4407:
4383:
4342:
4320:
4249:
4207:
4167:
4166:{\displaystyle b=a/2,}
4131:
4108:
3917:
3843:
3792:
3785:
3708:
3642:
3309:
3203:
2611:; treating them using
2501:is a vector space for
2487:
2422:
2372:
2326:
2245:
2205:
2104:, which is called the
1969:by a scalar belong to
1880:
1834:
1775:
1723:
1602:
1500:
1467:
1441:
1407:
1358:
1321:
1282:
1251:
1250:{\displaystyle s\in F}
1223:
622:A vector space over a
234:. This is the case of
195:if its dimension is a
75:
19734:Row and column spaces
19679:Scalar multiplication
19264:John Wiley & Sons
19242:John Wiley & Sons
19212:Differential geometry
19185:Jost, Jürgen (2005),
19147:John Wiley & Sons
18685:, Royal Irish Academy
18481:Historical references
18275:John Wiley & Sons
17738:Joshi, K. D. (1989),
17351:, Th. XIII.6, p. 349.
17127:, Th. 14.3. See also
16815:, ch. XII.3., p. 335.
16485:Stoll & Wong 1968
16436:, §1.1, Definition 2.
16265:
16230:
16188:
16186:{\displaystyle L^{2}}
16161:
16159:{\displaystyle L^{2}}
16113:
16006:
15945:This axiom is not an
15910:
15806:
15772:
15674:
15576:
15218:
15154:
15119:
15038:
14991:
14939:
14916:
14845:
14825:
14783:
14756:of two matrices, and
14747:
14694:
14674:
14646:
14530:
14474:
14454:
14434:
14377:
14323:
14298:
14228:
14196:
14159:differential operator
14108:
14075:
14044:
14008:
13981:
13979:{\displaystyle f_{n}}
13951:
13949:{\displaystyle g(x),}
13915:
13877:
13783:
13736:
13704:
13586:
13548:
13519:
13385:
13336:
13261:
13216:
13174:
13145:
13024:
12988:
12907:
12867:
12770:
12741:
12708:
12685:
12655:
12653:{\displaystyle 2^{n}}
12628:
12514:
12491:
12462:
12348:
12257:
12218:
12197:
12113:
12053:
12031:
12003:
12001:{\displaystyle V^{*}}
11974:
11937:
11905:
11885:
11861:
11792:
11770:
11741:
11701:
11665:
11618:
11595:
11558:
11537:
11509:
11497:pointwise convergence
11482:
11456:
11454:{\displaystyle f_{i}}
11429:
11406:
11356:
11266:
11240:
11220:
11192:
11162:
11142:
11122:
11102:
11080:
11046:
11013:
10876:corresponding to time
10869:
10814:
10772:
10639:
10602:
10560:
10446:
10421:
10399:
10367:
10256:
10254:{\displaystyle F^{n}}
10217:
10154:
10114:
10056:
10054:{\displaystyle f^{-}}
10029:
10009:
10007:{\displaystyle f^{+}}
9982:
9934:, are fundamental to
9928:Ordered vector spaces
9921:
9892:
9872:
9809:
9743:
9720:
9696:
9673:
9628:
9604:
9574:
9535:
9509:
9483:
9463:
9112:subject to the rules
9107:
8991:
8961:
8951:
8926:
8879:
8857:
8832:
8810:
8786:
8757:
8715:
8695:
8680:of two vector spaces
8675:
8646:
8593:
8573:
8526:
8480:
8478:{\displaystyle V_{i}}
8453:
8424:
8401:
8381:
8331:
8329:{\displaystyle V_{i}}
8304:
8244:Direct sum of modules
8221:
8141:
8102:
8063:
8043:
7985:
7920:
7874:
7850:
7777:
7744:
7721:
7719:{\displaystyle a_{i}}
7694:
7522:
7502:
7469:
7447:
7427:
7391:
7371:
7351:
7284:. The kernel and the
7279:
7259:
7237:
7215:
7183:
7145:
7125:
7105:
7076:
7044:
6988:
6922:
6862:
6827:
6792:
6772:
6750:
6683:
6661:
6641:
6618:, the quotient space
6613:
6592:. Given any subspace
6574:
6525:
6501:
6481:
6457:
6437:
6417:
6393:
6369:
6349:
6327:
6307:
6268:
6261:Quotient vector space
6221:Jordan canonical form
6163:
6074:is an element of the
6019:. Any nonzero vector
5891:
5848:
5810:
5788:
5771:matrix multiplication
5760:
5740:
5683:
5630:
5583:
5495:
5460:
5373:completely classified
5345:1-to-1 correspondence
5176:, the arrow going by
5111:
5099:, and vice versa via
4964:
4941:
4921:
4898:
4876:
4854:
4714:linear transformation
4685:
4683:{\displaystyle e^{x}}
4658:
4638:
4618:
4543:
4452:
4408:
4384:
4343:
4321:
4250:
4208:
4168:
4132:
4109:
3918:
3844:
3842:{\displaystyle (f+g)}
3786:
3709:
3671:
3643:
3310:
3244:(sequences of length
3213:Cartesian coordinates
3204:
2700:-integrable functions
2690:and the new field of
2599:and the inception of
2488:
2486:{\displaystyle F^{n}}
2423:
2373:
2327:
2246:
2206:
1881:
1835:
1776:
1724:
1528:
1501:
1468:
1442:
1408:
1359:
1322:
1283:
1252:
1224:
719:scalar multiplication
711:of these two vectors.
191:). A vector space is
127:complex vector spaces
112:scalar multiplication
47:
19869:Gram–Schmidt process
19821:Gaussian elimination
19218:, pp. xiv+352,
19091:, Berlin, New York:
18931:, Berlin, New York:
18906:, Berlin, New York:
18871:Axiomatic set theory
18848:Macdonald, Ian Grant
18711:Historia Mathematica
18662:, World Scientific,
18578:Historia Mathematica
18331:, Berlin, New York:
18155:, Providence, R.I.:
18131:The Calculus Gallery
18042:, Berlin, New York:
18020:, Berlin, New York:
17767:Lang, Serge (1987),
17718:Jain, M. C. (2001),
17423:, ch. III.1, p. 121.
17279:, Cor. 4.1.2, p. 69.
16971:, ch. 8, p. 135–156.
16292:That is, there is a
16238:
16210:
16170:
16143:
16120:Minkowski inequality
16045:
15987:
15981:equivalence relation
15947:associative property
15930:inner product spaces
15887:
15785:
15748:
15609:
15553:locally free modules
15349:). For example, the
15223:such that for every
15195:
15051:
15000:
14956:
14928:
14857:
14834:
14802:
14760:
14703:
14683:
14663:
14545:
14486:
14463:
14443:
14411:
14375:{\displaystyle p(t)}
14357:
14337:algebra over a field
14307:
14241:
14205:
14185:Algebra over a field
14179:Algebras over fields
14170:decomposes a linear
14139:Schrödinger equation
14119:Gram–Schmidt process
14094:
14061:
14021:
13997:
13988:Taylor approximation
13963:
13928:
13886:
13792:
13753:
13749:. The Hilbert space
13595:
13557:
13546:{\displaystyle f(x)}
13528:
13394:
13345:
13280:
13229:
13183:
13163:
13154:integrable functions
13036:
12999:
12876:
12778:
12768:{\displaystyle p=1:}
12750:
12721:
12694:
12664:
12637:
12523:
12500:
12473:
12356:
12266:
12228:
12207:
12123:
12095:
12040:
12012:
11985:
11954:
11914:
11894:
11874:
11801:
11779:
11750:
11718:
11678:
11639:
11604:
11569:
11547:
11517:
11468:
11438:
11415:
11369:
11263:
11229:
11201:
11171:
11151:
11131:
11111:
11089:
11067:
11035:
10886:
10823:
10785:
10648:
10619:
10569:
10459:
10430:
10408:
10376:
10269:
10238:
10229:inner product spaces
10225:normed vector spaces
10165:
10124:
10091:
10038:
10018:
9991:
9942:
9936:Lebesgue integration
9901:
9881:
9856:
9751:
9729:
9709:
9682:
9641:
9617:
9583:
9547:
9518:
9492:
9472:
9116:
9003:
8977:
8935:
8888:
8866:
8841:
8819:
8799:
8769:
8728:
8704:
8684:
8655:
8619:
8582:
8539:
8492:
8462:
8433:
8413:
8390:
8340:
8313:
8266:
8170:
8164:rank–nullity theorem
8130:
8126:(over a fixed field
8076:
8052:
7994:
7929:
7883:
7863:
7753:
7730:
7703:
7531:
7511:
7478:
7456:
7436:
7403:
7380:
7360:
7291:
7268:
7246:
7224:
7220:consists of vectors
7192:
7160:
7134:
7114:
7085:
7056:
6997:
6931:
6871:
6836:
6801:
6781:
6759:
6692:
6672:
6650:
6622:
6596:
6557:
6514:
6490:
6470:
6446:
6426:
6406:
6382:
6358:
6336:
6316:
6296:
6197:algebraically closed
6125:
5821:
5797:
5777:
5749:
5516:
5484:
5285:. Via the injective
5057:is both one-to-one (
4950:
4930:
4907:
4885:
4863:
4724:
4667:
4647:
4627:
4554:
4467:
4421:
4392:
4352:
4330:
4261:
4223:
4177:
4140:
4118:
3992:
3853:
3821:
3718:
3676:
3614:
3254:
3009:
2570:equivalence relation
2470:
2386:
2336:
2255:
2227:
2144:
1912:linearly independent
1844:
1785:
1733:
1635:
1477:
1451:
1418:
1369:
1332:
1295:
1261:
1235:
1175:
1124:complex vector space
730:and any vector
653:are commonly called
499:Group with operators
442:Complemented lattice
277:Algebraic structures
256:inner product spaces
244:associative algebras
242:, polynomial rings,
201:infinite-dimensional
20055:Concepts in physics
19999:Numerical stability
19879:Multilinear algebra
19854:Inner product space
19704:Linear independence
19060:Griffiths, David J.
19009:Commutative algebra
18954:Projective Geometry
18792:Solid State Physics
18739:(in Italian), Turin
18431:Schaefer, Helmut H.
18414:Functional analysis
17103:, ch. 1, pp. 31–32.
16703:, pp. 268–271.
16139:"Many functions in
16118:is provided by the
16039:triangle inequality
16004:{\displaystyle V/W}
15497:instead of a field
15286:are required to be
14394:, because they are
14053:is commonly called
12633:in which the first
12089:A first example is
12060:Hahn–Banach theorem
12058:). The fundamental
11868:functional analysis
11501:uniform convergence
11493:mode of convergence
11057:close to each other
10075:Inner product space
10071:Normed vector space
10061:the negative part.
9843:functional analysis
9814:This is called the
8963:Commutative diagram
8722:multilinear algebra
7242:that are mapped to
6639:{\displaystyle V/W}
6572:{\displaystyle n-1}
6508:linear combinations
6486:containing the set
6285:(green and yellow).
6243:Basic constructions
5892:The volume of this
5512:, by the following
5118:by its coordinates
3960:functional analysis
2721:Arrows in the plane
2692:functional analysis
2665:Salvatore Pincherle
2644:linear independence
2613:linear combinations
2584:of that relation.
1894:Linear independence
1466:{\displaystyle s=0}
553:Composition algebra
313:Quasigroup and loop
199:. Otherwise, it is
150:physical quantities
19709:Linear combination
19538:Weibel, Charles A.
19493:(1): xviii–xxxiv,
19389:10.1007/bf02242355
19344:Dover Publications
19308:Misner, Charles W.
19281:Mac Lane, Saunders
19216:Dover Publications
18877:, pp. 31–33,
18774:Further references
18622:Grassmann, Hermann
17827:Mac Lane, Saunders
17375:, Lemma III.16.11.
16460:, pp. 99–101.
16260:
16225:
16183:
16156:
16108:
16001:
15905:
15801:
15767:
15691:by a fixed vector
15669:
15591:
15509:-module (that is,
15413:hairy ball theorem
15268:makes the product
15213:
15170:
15132:Related structures
15114:
15047:, whereas forcing
15033:
14986:
14934:
14911:
14840:
14820:
14778:
14742:
14689:
14669:
14641:
14528:{\displaystyle =-}
14525:
14469:
14449:
14429:
14392:algebraic geometry
14390:form the basis of
14372:
14329:
14318:
14293:
14223:
14106:{\displaystyle H,}
14103:
14073:{\displaystyle H,}
14070:
14039:
14003:
13976:
13946:
13910:
13872:
13778:
13739:
13699:
13613:
13581:
13543:
13514:
13421:
13380:
13331:
13256:
13211:
13169:
13140:
13019:
12983:
12862:
12765:
12736:
12706:{\displaystyle 0,}
12703:
12680:
12650:
12623:
12512:{\displaystyle p.}
12509:
12486:
12457:
12395:
12343:
12299:
12252:
12213:
12192:
12108:
12048:
12026:
11998:
11969:
11932:
11900:
11880:
11856:
11819:
11787:
11765:
11736:
11696:
11670:equipped with the
11660:
11625:
11613:
11590:
11553:
11532:
11480:{\displaystyle V,}
11477:
11451:
11427:{\displaystyle V.}
11424:
11401:
11351:
11321:
11235:
11215:
11187:
11157:
11137:
11117:
11097:
11075:
11041:
11008:
10880:special relativity
10864:
10809:
10767:
10634:
10597:
10555:
10441:
10416:
10394:
10362:
10251:
10212:
10149:
10109:
10051:
10024:
10004:
9977:
9916:
9887:
9867:
9816:universal property
9804:
9741:{\displaystyle g:}
9738:
9715:
9694:{\displaystyle u,}
9691:
9668:
9623:
9599:
9569:
9530:
9504:
9478:
9458:
9456:
9102:
8989:{\displaystyle g,}
8986:
8967:
8946:
8921:
8874:
8852:
8827:
8805:
8781:
8752:
8710:
8690:
8670:
8641:
8588:
8568:
8557:
8521:
8510:
8475:
8448:
8419:
8396:
8376:
8326:
8299:
8298:
8286:
8216:
8136:
8097:
8058:
8038:
7980:
7915:
7869:
7845:
7742:{\displaystyle x,}
7739:
7716:
7689:
7517:
7497:
7464:
7442:
7422:
7386:
7366:
7346:
7274:
7254:
7232:
7210:
7178:
7140:
7120:
7100:
7071:
7052:the difference of
7039:
6983:
6917:
6857:
6822:
6787:
6767:
6745:
6678:
6656:
6636:
6608:
6569:
6520:
6496:
6476:
6452:
6432:
6412:
6388:
6364:
6344:
6322:
6312:of a vector space
6302:
6287:
6158:
6115:having eigenvalue
6078:of the difference
5925:
5843:
5805:
5783:
5769:, or by using the
5758:{\textstyle \sum }
5755:
5735:
5490:
5463:
5188:is negative), and
5132:
4962:{\displaystyle F.}
4959:
4936:
4919:{\displaystyle V,}
4916:
4893:
4871:
4849:
4847:
4680:
4653:
4633:
4613:
4538:
4447:
4403:
4379:
4338:
4316:
4310:
4245:
4203:
4163:
4130:{\displaystyle a,}
4127:
4104:
4102:
3913:
3839:
3793:
3781:
3704:
3638:
3305:
3199:
3197:
2483:
2418:
2368:
2322:
2241:
2211:of a vector space
2201:
2002:of a vector space
1949:of a vector space
1876:
1830:
1771:
1719:
1607:Linear combination
1603:
1496:
1463:
1437:
1403:
1354:
1317:
1278:
1247:
1219:
1142:vector space over
925:of vector addition
874:of vector addition
847:of vector addition
812:of vector addition
740:another vector in
693:a third vector in
193:finite-dimensional
123:Real vector spaces
76:
20040:
20039:
19907:Geometric algebra
19864:Kronecker product
19699:Linear projection
19684:Vector projection
19551:978-0-521-55987-4
19509:10.1109/30.125072
19471:978-0-13-535732-3
19449:978-0-412-10800-6
19353:978-0-486-43235-9
19331:978-0-7167-0344-0
19324:, W. H. Freeman,
19300:978-0-387-98403-2
19273:978-0-471-18117-0
19251:978-0-471-15496-9
19225:978-0-486-66721-8
19200:978-3-540-25907-7
19178:978-0-387-94087-8
19102:978-0-387-90093-3
19077:978-0-13-124405-4
19052:978-0-412-60610-6
19022:978-0-387-94269-8
18967:978-0-387-96532-1
18942:978-3-540-64241-1
18925:Bourbaki, Nicolas
18917:978-3-540-64243-5
18900:Bourbaki, Nicolas
18884:978-0-8218-5026-8
18828:978-0-201-09394-0
18803:978-0-03-083993-1
18757:Peano, G. (1901)
18669:978-981-12-4103-1
18649:978-0-8218-2031-5
18560:Bourbaki, Nicolas
18543:Bellavitis, Giuso
18448:978-1-4612-7155-0
18370:978-0-486-48123-4
18360:2027/uc1.b4250788
18342:978-0-387-94001-4
18317:978-0-201-14179-5
18284:978-0-471-50459-7
18259:978-0-471-85824-9
18238:978-0-88385-031-2
18220:978-0-201-59619-9
18202:978-0-387-98485-8
18184:978-0-534-17094-3
18166:978-0-8218-0772-9
18144:978-0-691-09565-3
18122:978-0-534-10050-6
18104:978-0-486-69193-0
18053:978-0-387-97894-9
18031:978-3-540-41129-1
18014:Bourbaki, Nicolas
18006:978-3-540-13627-9
17989:Bourbaki, Nicolas
17972:978-3-540-56799-8
17938:978-0-8176-4840-4
17918:978-0-8247-9144-5
17900:978-0-387-24766-3
17864:978-0-89871-454-8
17840:978-0-8218-1646-2
17811:978-0-387-95385-4
17786:978-1-4757-1949-9
17760:978-1-119-45592-9
17731:978-0-8493-0978-6
17711:978-0-8176-4687-5
17673:978-0-387-71568-1
17647:978-0-8247-8419-5
17625:978-0-89871-510-1
16875:, Example IV.2.6.
16515:, p. 10–11;
16503:, pp. 41–42.
16329:Husemoller (1994)
16124:almost everywhere
16022:uniform structure
15899:
15429:division algebras
15254:, the projection
15178:topological space
15045:symmetric algebra
14937:{\displaystyle n}
14843:{\displaystyle V}
14788:endowed with the
14692:{\displaystyle n}
14672:{\displaystyle n}
14537:anticommutativity
14472:{\displaystyle y}
14452:{\displaystyle x}
14333:bilinear operator
14143:quantum mechanics
14055:Fourier expansion
14006:{\displaystyle f}
13922:complex conjugate
13908:
13860:
13809:
13803:
13598:
13410:
13397:
13134:
13031:Lebesgue integral
12746:but does not for
12713:converges to the
12443:
12436:
12386:
12341:
12324:
12290:
12216:{\displaystyle p}
12091:the vector space
11883:{\displaystyle 1}
11804:
11663:{\displaystyle ,}
11556:{\displaystyle p}
11434:For example, the
11306:
11305:
11299:
11252:one can consider
11238:{\displaystyle F}
11160:{\displaystyle F}
11140:{\displaystyle a}
11120:{\displaystyle V}
11044:{\displaystyle V}
10779:positive definite
10234:Coordinate space
10207:
10027:{\displaystyle f}
9890:{\displaystyle n}
9718:{\displaystyle f}
9626:{\displaystyle X}
9481:{\displaystyle f}
9405:
9395:
9302:
9292:
9243:
9242: is a scalar
9235:
9234: where
9231:
9228:
9194:
9188:
9160:
9150:
8808:{\displaystyle g}
8764:Cartesian product
8713:{\displaystyle W}
8693:{\displaystyle V}
8591:{\displaystyle I}
8542:
8495:
8422:{\displaystyle I}
8399:{\displaystyle i}
8271:
8139:{\displaystyle F}
8061:{\displaystyle c}
7872:{\displaystyle f}
7840:
7726:are functions in
7678:
7625:
7578:
7520:{\displaystyle A}
7445:{\displaystyle A}
7396:, respectively.
7389:{\displaystyle W}
7369:{\displaystyle V}
7356:are subspaces of
7277:{\displaystyle W}
7143:{\displaystyle W}
7123:{\displaystyle W}
6790:{\displaystyle V}
6681:{\displaystyle W}
6659:{\displaystyle V}
6545:respectively. If
6523:{\displaystyle S}
6499:{\displaystyle S}
6479:{\displaystyle V}
6455:{\displaystyle S}
6435:{\displaystyle V}
6415:{\displaystyle V}
6391:{\displaystyle V}
6367:{\displaystyle V}
6325:{\displaystyle V}
6305:{\displaystyle W}
6273:(blue, thick) in
6121:is equivalent to
6089:(where Id is the
5786:{\displaystyle A}
5493:{\displaystyle A}
5395:is isomorphic to
5276:dual vector space
5150:coordinate system
5144:departing at the
4939:{\displaystyle a}
4656:{\displaystyle b}
4636:{\displaystyle a}
3955:differentiability
3633:
3503:for real numbers
3215:of its endpoint.
2582:equivalence class
2566:Bellavitis (1833)
2546:analytic geometry
2458:coordinate vector
2140:Consider a basis
2125:, depends on the
1914:if no element of
1627:is an element of
1615:of elements of a
1166:endomorphism ring
1156:ring homomorphism
1116:real vector space
1108:
1107:
707:, and called the
612:
611:
205:infinite cardinal
146:Euclidean vectors
16:(Redirected from
20082:
20030:
20029:
19912:Exterior algebra
19849:Hadamard product
19766:
19754:Linear equations
19625:
19618:
19611:
19602:
19601:
19597:
19571:
19533:
19532:
19531:
19525:
19519:, archived from
19502:
19484:
19474:
19452:
19440:Chapman and Hall
19436:Chapman and Hall
19433:
19419:
19407:
19383:(3–4): 281–292,
19373:Strassen, Volker
19364:
19334:
19303:
19276:
19254:
19239:
19228:
19203:
19181:
19159:
19156:978-0470-88861-2
19137:
19128:
19105:
19080:
19055:
19043:Chapman and Hall
19033:
19000:
18970:
18945:
18920:
18895:
18868:
18858:
18839:
18806:
18795:
18786:Mermin, N. David
18767:Internet Archive
18754:
18748:
18740:
18728:
18727:
18704:
18686:
18672:
18652:
18636:Extension Theory
18631:
18617:
18603:
18594:
18567:
18554:
18538:
18529:Bolzano, Bernard
18524:
18497:
18475:
18465:Treves, François
18460:
18426:
18408:
18381:
18362:
18345:
18320:
18295:
18262:
18241:
18223:
18205:
18187:
18169:
18147:
18125:
18107:
18089:
18079:Choquet, Gustave
18074:
18056:
18034:
18009:
17975:
17950:
17949:, Academic Press
17941:
17921:
17903:
17878:
17867:
17843:
17822:
17789:
17763:
17743:
17734:
17714:
17688:
17676:
17656:Abstract algebra
17650:
17639:
17628:
17603:
17580:
17574:
17568:
17562:
17556:
17550:
17544:
17538:
17532:
17526:
17520:
17514:
17508:
17502:
17496:
17490:
17484:
17478:
17472:
17466:
17460:
17454:
17448:
17445:Varadarajan 1974
17442:
17436:
17430:
17424:
17418:
17412:
17406:
17400:
17394:
17388:
17382:
17376:
17370:
17364:
17358:
17352:
17346:
17340:
17334:
17328:
17322:
17316:
17310:
17304:
17298:
17292:
17286:
17280:
17274:
17268:
17262:
17256:
17250:
17244:
17238:
17232:
17226:
17220:
17214:
17208:
17198:
17192:
17186:
17180:
17174:
17168:
17162:
17156:
17150:
17144:
17138:
17132:
17122:
17116:
17110:
17104:
17098:
17092:
17086:
17080:
17074:
17068:
17062:
17056:
17050:
17044:
17038:
17032:
17026:
17020:
17014:
17008:
17002:
16996:
16995:, ch. 8, p. 140.
16990:
16984:
16978:
16972:
16966:
16960:
16954:
16948:
16942:
16936:
16930:
16924:
16918:
16912:
16906:
16900:
16894:
16888:
16882:
16876:
16870:
16864:
16858:
16852:
16846:
16840:
16834:
16828:
16822:
16816:
16810:
16804:
16798:
16792:
16786:
16780:
16770:
16764:
16754:
16748:
16738:
16732:
16722:
16716:
16710:
16704:
16698:
16692:
16686:
16680:
16674:
16668:
16662:
16656:
16650:
16644:
16638:
16632:
16626:
16620:
16614:
16608:
16602:
16596:
16590:
16584:
16574:
16568:
16558:
16552:
16542:
16536:
16530:
16524:
16510:
16504:
16498:
16492:
16482:
16473:
16467:
16461:
16455:
16449:
16443:
16437:
16431:
16425:
16419:
16413:
16399:
16393:
16387:
16381:
16375:
16369:
16363:
16357:
16351:
16336:
16317:
16311:
16309:
16290:
16284:
16277:
16271:
16269:
16267:
16266:
16261:
16250:
16249:
16234:
16232:
16231:
16226:
16204:
16198:
16192:
16190:
16189:
16184:
16182:
16181:
16165:
16163:
16162:
16157:
16155:
16154:
16137:
16131:
16117:
16115:
16114:
16109:
16107:
16106:
16088:
16087:
16069:
16068:
16035:
16029:
16018:
16012:
16010:
16008:
16007:
16002:
15997:
15973:
15967:
15956:
15950:
15943:
15937:
15922:
15916:
15914:
15912:
15911:
15906:
15901:
15900:
15892:
15881:
15846:projective space
15820:
15810:
15808:
15807:
15802:
15800:
15792:
15776:
15774:
15773:
15768:
15766:
15758:
15743:
15732:
15710:
15700:
15678:
15676:
15675:
15670:
15665:
15657:
15646:
15638:
15581:(light blue) in
15569:Projective space
15458:cotangent bundle
15402:
15373:
15348:
15334:
15277:
15267:
15245:
15222:
15220:
15219:
15214:
15167:
15126:exterior algebra
15123:
15121:
15120:
15115:
15113:
15112:
15107:
15098:
15097:
15092:
15080:
15079:
15074:
15065:
15064:
15059:
15042:
15040:
15039:
15034:
15029:
15028:
15023:
15014:
15013:
15008:
14995:
14993:
14992:
14987:
14985:
14984:
14979:
14970:
14969:
14964:
14946:distributive law
14943:
14941:
14940:
14935:
14920:
14918:
14917:
14912:
14907:
14906:
14901:
14886:
14885:
14880:
14871:
14870:
14865:
14849:
14847:
14846:
14841:
14829:
14827:
14826:
14821:
14787:
14785:
14784:
14779:
14774:
14773:
14768:
14751:
14749:
14748:
14743:
14698:
14696:
14695:
14690:
14678:
14676:
14675:
14670:
14650:
14648:
14647:
14642:
14534:
14532:
14531:
14526:
14478:
14476:
14475:
14470:
14458:
14456:
14455:
14450:
14438:
14436:
14435:
14432:{\displaystyle }
14430:
14381:
14379:
14378:
14373:
14327:
14325:
14324:
14319:
14314:
14302:
14300:
14299:
14294:
14268:
14248:
14232:
14230:
14229:
14224:
14175:
14172:compact operator
14168:spectral theorem
14112:
14110:
14109:
14104:
14079:
14077:
14076:
14071:
14048:
14046:
14045:
14042:{\displaystyle }
14040:
14012:
14010:
14009:
14004:
13985:
13983:
13982:
13977:
13975:
13974:
13955:
13953:
13952:
13947:
13919:
13917:
13916:
13911:
13909:
13904:
13890:
13881:
13879:
13878:
13873:
13861:
13856:
13842:
13828:
13827:
13807:
13801:
13787:
13785:
13784:
13779:
13765:
13764:
13708:
13706:
13705:
13700:
13692:
13674:
13673:
13668:
13664:
13654:
13653:
13623:
13622:
13612:
13590:
13588:
13587:
13582:
13571:
13570:
13552:
13550:
13549:
13544:
13523:
13521:
13520:
13515:
13507:
13489:
13488:
13483:
13479:
13469:
13468:
13447:
13446:
13431:
13430:
13420:
13408:
13389:
13387:
13386:
13381:
13370:
13369:
13360:
13359:
13340:
13338:
13337:
13332:
13324:
13323:
13305:
13304:
13292:
13291:
13270:Riemann integral
13265:
13263:
13262:
13257:
13243:
13242:
13220:
13218:
13217:
13212:
13201:
13200:
13178:
13176:
13175:
13170:
13149:
13147:
13146:
13141:
13136:
13135:
13127:
13125:
13121:
13120:
13102:
13101:
13096:
13078:
13073:
13072:
13054:
13053:
13028:
13026:
13025:
13020:
13018:
12992:
12990:
12989:
12984:
12976:
12975:
12960:
12959:
12947:
12946:
12933:
12932:
12931:
12921:
12903:
12902:
12893:
12892:
12887:
12871:
12869:
12868:
12863:
12852:
12851:
12827:
12826:
12805:
12804:
12795:
12794:
12789:
12774:
12772:
12771:
12766:
12745:
12743:
12742:
12737:
12712:
12710:
12709:
12704:
12689:
12687:
12686:
12681:
12679:
12678:
12659:
12657:
12656:
12651:
12649:
12648:
12632:
12630:
12629:
12624:
12619:
12615:
12596:
12595:
12574:
12573:
12558:
12557:
12537:
12536:
12531:
12518:
12516:
12515:
12510:
12495:
12493:
12492:
12487:
12485:
12484:
12466:
12464:
12463:
12458:
12444:
12441:
12438:
12437:
12429:
12427:
12423:
12422:
12421:
12416:
12410:
12409:
12400:
12394:
12376:
12375:
12366:
12352:
12350:
12349:
12344:
12342:
12339:
12325:
12322:
12319:
12314:
12313:
12304:
12298:
12286:
12285:
12276:
12261:
12259:
12258:
12253:
12222:
12220:
12219:
12214:
12201:
12199:
12198:
12193:
12191:
12187:
12180:
12179:
12161:
12160:
12148:
12147:
12130:
12117:
12115:
12114:
12109:
12107:
12106:
12082:, introduced by
12063:
12057:
12055:
12054:
12049:
12047:
12035:
12033:
12032:
12027:
12025:
12007:
12005:
12004:
11999:
11997:
11996:
11978:
11976:
11975:
11970:
11941:
11939:
11938:
11933:
11928:
11927:
11922:
11909:
11907:
11906:
11901:
11889:
11887:
11886:
11881:
11865:
11863:
11862:
11857:
11849:
11844:
11836:
11835:
11830:
11824:
11818:
11796:
11794:
11793:
11788:
11786:
11774:
11772:
11771:
11766:
11764:
11763:
11758:
11745:
11743:
11742:
11739:{\displaystyle }
11737:
11705:
11703:
11702:
11699:{\displaystyle }
11697:
11669:
11667:
11666:
11661:
11622:
11620:
11619:
11614:
11599:
11597:
11596:
11591:
11562:
11560:
11559:
11554:
11541:
11539:
11538:
11533:
11531:
11530:
11525:
11486:
11484:
11483:
11478:
11460:
11458:
11457:
11452:
11450:
11449:
11433:
11431:
11430:
11425:
11410:
11408:
11407:
11402:
11394:
11393:
11381:
11380:
11360:
11358:
11357:
11352:
11350:
11349:
11331:
11330:
11320:
11303:
11297:
11296:
11295:
11285:
11280:
11256:of vectors. The
11244:
11242:
11241:
11236:
11224:
11222:
11221:
11216:
11211:
11196:
11194:
11193:
11188:
11186:
11178:
11166:
11164:
11163:
11158:
11146:
11144:
11143:
11138:
11126:
11124:
11123:
11118:
11106:
11104:
11103:
11098:
11096:
11084:
11082:
11081:
11076:
11074:
11050:
11048:
11047:
11042:
11017:
11015:
11014:
11009:
11004:
11003:
10994:
10993:
10981:
10980:
10971:
10970:
10958:
10957:
10948:
10947:
10935:
10934:
10925:
10924:
10906:
10901:
10896:
10873:
10871:
10870:
10865:
10830:
10818:
10816:
10815:
10810:
10805:
10800:
10795:
10776:
10774:
10773:
10768:
10763:
10762:
10753:
10752:
10740:
10739:
10730:
10729:
10717:
10716:
10707:
10706:
10694:
10693:
10684:
10683:
10668:
10663:
10658:
10643:
10641:
10640:
10635:
10633:
10632:
10627:
10606:
10604:
10603:
10598:
10587:
10579:
10564:
10562:
10561:
10556:
10551:
10546:
10541:
10533:
10528:
10523:
10515:
10511:
10507:
10499:
10474:
10466:
10450:
10448:
10447:
10442:
10437:
10425:
10423:
10422:
10417:
10415:
10403:
10401:
10400:
10395:
10390:
10389:
10384:
10371:
10369:
10368:
10363:
10358:
10357:
10348:
10347:
10329:
10328:
10319:
10318:
10306:
10298:
10287:
10279:
10260:
10258:
10257:
10252:
10250:
10249:
10231:, respectively.
10222:
10221:
10219:
10218:
10213:
10208:
10203:
10195:
10187:
10182:
10177:
10172:
10159:
10158:
10156:
10155:
10150:
10142:
10134:
10118:
10116:
10115:
10110:
10108:
10103:
10098:
10060:
10058:
10057:
10052:
10050:
10049:
10033:
10031:
10030:
10025:
10013:
10011:
10010:
10005:
10003:
10002:
9986:
9984:
9983:
9978:
9973:
9972:
9960:
9959:
9925:
9923:
9922:
9917:
9915:
9914:
9909:
9896:
9894:
9893:
9888:
9876:
9874:
9873:
9868:
9846:
9813:
9811:
9810:
9805:
9797:
9789:
9772:
9764:
9747:
9745:
9744:
9739:
9724:
9722:
9721:
9716:
9700:
9698:
9697:
9692:
9677:
9675:
9674:
9669:
9632:
9630:
9629:
9624:
9608:
9606:
9605:
9600:
9598:
9590:
9578:
9576:
9575:
9570:
9565:
9557:
9539:
9537:
9536:
9531:
9513:
9511:
9510:
9505:
9487:
9485:
9484:
9479:
9467:
9465:
9464:
9459:
9457:
9454:
9453:
9448:
9447:
9442:
9433:
9425:
9424:
9419:
9410:
9403:
9393:
9389:
9388:
9383:
9374:
9373:
9368:
9356:
9348:
9347:
9345:
9337:
9336:
9331:
9322:
9314:
9313:
9308:
9300:
9290:
9289:
9278:
9277:
9272:
9263:
9262:
9257:
9244:
9241:
9236:
9233:
9229:
9226:
9224:
9216:
9199:
9192:
9186:
9185:
9174:
9158:
9148:
9144:
9136:
9111:
9109:
9108:
9103:
9098:
9097:
9092:
9083:
9082:
9077:
9062:
9061:
9056:
9047:
9046:
9041:
9032:
9031:
9026:
9017:
9016:
9011:
8995:
8993:
8992:
8987:
8955:
8953:
8952:
8947:
8942:
8930:
8928:
8927:
8922:
8917:
8909:
8895:
8883:
8881:
8880:
8875:
8873:
8861:
8859:
8858:
8853:
8848:
8836:
8834:
8833:
8828:
8826:
8814:
8812:
8811:
8806:
8790:
8788:
8787:
8782:
8761:
8759:
8758:
8753:
8719:
8717:
8716:
8711:
8699:
8697:
8696:
8691:
8679:
8677:
8676:
8671:
8650:
8648:
8647:
8642:
8634:
8633:
8597:
8595:
8594:
8589:
8577:
8575:
8574:
8569:
8567:
8566:
8556:
8530:
8528:
8527:
8522:
8520:
8519:
8509:
8484:
8482:
8481:
8476:
8474:
8473:
8457:
8455:
8454:
8449:
8447:
8446:
8441:
8428:
8426:
8425:
8420:
8405:
8403:
8402:
8397:
8385:
8383:
8382:
8377:
8375:
8374:
8363:
8359:
8358:
8353:
8335:
8333:
8332:
8327:
8325:
8324:
8308:
8306:
8305:
8300:
8297:
8296:
8295:
8285:
8225:
8223:
8222:
8217:
8180:
8148:abelian category
8145:
8143:
8142:
8137:
8118:
8112:
8106:
8104:
8103:
8098:
8067:
8065:
8064:
8059:
8047:
8045:
8044:
8039:
8037:
8036:
8018:
8017:
7989:
7987:
7986:
7981:
7979:
7978:
7966:
7965:
7953:
7952:
7924:
7922:
7921:
7916:
7914:
7913:
7898:
7897:
7878:
7876:
7875:
7870:
7859:of the function
7854:
7852:
7851:
7846:
7841:
7839:
7838:
7837:
7824:
7820:
7819:
7809:
7807:
7806:
7796:
7791:
7748:
7746:
7745:
7740:
7725:
7723:
7722:
7717:
7715:
7714:
7698:
7696:
7695:
7690:
7679:
7677:
7676:
7675:
7662:
7658:
7657:
7647:
7645:
7644:
7626:
7624:
7623:
7622:
7609:
7605:
7604:
7594:
7592:
7591:
7579:
7577:
7569:
7561:
7559:
7558:
7543:
7542:
7526:
7524:
7523:
7518:
7506:
7504:
7503:
7498:
7496:
7488:
7473:
7471:
7470:
7465:
7463:
7451:
7449:
7448:
7443:
7431:
7429:
7428:
7423:
7421:
7410:
7395:
7393:
7392:
7387:
7375:
7373:
7372:
7367:
7355:
7353:
7352:
7347:
7336:
7325:
7283:
7281:
7280:
7275:
7263:
7261:
7260:
7255:
7253:
7241:
7239:
7238:
7233:
7231:
7219:
7217:
7216:
7211:
7188:of a linear map
7187:
7185:
7184:
7179:
7149:
7147:
7146:
7141:
7129:
7127:
7126:
7121:
7109:
7107:
7106:
7101:
7099:
7098:
7093:
7080:
7078:
7077:
7072:
7070:
7069:
7064:
7048:
7046:
7045:
7040:
7032:
7031:
7026:
7011:
7010:
7005:
6992:
6990:
6989:
6984:
6973:
6947:
6926:
6924:
6923:
6918:
6910:
6906:
6905:
6904:
6899:
6890:
6889:
6884:
6866:
6864:
6863:
6858:
6850:
6849:
6844:
6831:
6829:
6828:
6823:
6815:
6814:
6809:
6796:
6794:
6793:
6788:
6776:
6774:
6773:
6768:
6766:
6754:
6752:
6751:
6746:
6732:
6724:
6716:
6699:
6687:
6685:
6684:
6679:
6665:
6663:
6662:
6657:
6645:
6643:
6642:
6637:
6632:
6617:
6615:
6614:
6609:
6578:
6576:
6575:
6570:
6529:
6527:
6526:
6521:
6505:
6503:
6502:
6497:
6485:
6483:
6482:
6477:
6461:
6459:
6458:
6453:
6441:
6439:
6438:
6433:
6421:
6419:
6418:
6413:
6397:
6395:
6394:
6389:
6373:
6371:
6370:
6365:
6353:
6351:
6350:
6345:
6343:
6331:
6329:
6328:
6323:
6311:
6309:
6308:
6303:
6280:
6238:
6228:
6214:
6208:
6195:
6189:
6183:
6173:
6167:
6165:
6164:
6159:
6120:
6114:
6108:
6102:
6088:
6073:
6068:. Equivalently,
6067:
6058:
6048:
6042:
6024:
6018:
6007:
6001:
5995:
5952:
5946:
5937:
5922:
5913:
5904:
5884:
5866:
5860:
5852:
5850:
5849:
5844:
5839:
5828:
5814:
5812:
5811:
5806:
5804:
5792:
5790:
5789:
5784:
5764:
5762:
5761:
5756:
5744:
5742:
5741:
5736:
5731:
5727:
5726:
5725:
5716:
5715:
5702:
5697:
5673:
5672:
5663:
5662:
5649:
5644:
5626:
5625:
5616:
5615:
5602:
5597:
5571:
5570:
5552:
5551:
5539:
5538:
5523:
5511:
5505:
5499:
5497:
5496:
5491:
5479:
5473:
5461:A typical matrix
5438:
5432:
5426:
5420:
5414:
5400:
5394:
5388:
5370:
5364:
5358:
5352:
5342:
5331:
5325:
5311:
5306:Once a basis of
5298:
5284:
5272:
5266:
5260:
5248:
5236:
5218:
5205:
5199:
5193:
5187:
5181:
5175:
5163:
5157:
5148:of some (fixed)
5143:
5129:
5123:
5117:
5104:
5098:
5092:
5086:
5076:
5070:
5056:
5051:. Equivalently,
5046:
5028:
5007:
4990:
4977:is a linear map
4968:
4966:
4965:
4960:
4945:
4943:
4942:
4937:
4925:
4923:
4922:
4917:
4902:
4900:
4899:
4894:
4892:
4880:
4878:
4877:
4872:
4870:
4858:
4856:
4855:
4850:
4848:
4841:
4814:
4787:
4770:
4749:
4741:
4689:
4687:
4686:
4681:
4679:
4678:
4662:
4660:
4659:
4654:
4642:
4640:
4639:
4634:
4622:
4620:
4619:
4614:
4609:
4608:
4587:
4586:
4547:
4545:
4544:
4539:
4507:
4506:
4482:
4481:
4456:
4454:
4453:
4448:
4428:
4412:
4410:
4409:
4404:
4402:
4388:
4386:
4385:
4380:
4347:
4345:
4344:
4339:
4337:
4325:
4323:
4322:
4317:
4315:
4314:
4254:
4252:
4251:
4246:
4241:
4233:
4212:
4210:
4209:
4204:
4199:
4172:
4170:
4169:
4164:
4156:
4136:
4134:
4133:
4128:
4113:
4111:
4110:
4105:
4103:
4059:
4052:
4006:
3999:
3998:
3966:Linear equations
3944:
3926:
3922:
3920:
3919:
3914:
3848:
3846:
3845:
3840:
3817:is the function
3816:
3810:
3804:
3798:
3790:
3788:
3787:
3782:
3713:
3711:
3710:
3705:
3703:
3695:
3653:
3647:
3645:
3644:
3639:
3634:
3629:
3621:
3609:
3603:
3597:
3591:
3582:
3571:field extensions
3569:More generally,
3561:
3549:
3532:
3526:
3520:
3514:
3508:
3502:
3467:
3423:
3417:
3411:
3402:
3392:
3371:
3364:
3354:
3348:
3340:coordinate space
3337:
3331:
3325:
3314:
3312:
3311:
3306:
3301:
3300:
3282:
3281:
3269:
3268:
3249:
3241:
3234:
3228:
3219:Coordinate space
3208:
3206:
3205:
3200:
3198:
3135:
3134:
3122:
3121:
3109:
3108:
3096:
3095:
3073:
3072:
3060:
3059:
3041:
3040:
3028:
3027:
3004:
2988:
2984:
2980:
2976:
2964:
2958:
2947:
2937:
2930:
2924:
2918:
2909:
2899:
2890:
2884:
2876:. It is denoted
2875:
2869:
2859:
2853:
2847:
2838:
2820:
2814:
2785:
2778:
2769:
2757:
2751:
2745:
2733:
2578:Euclidean vector
2542:Pierre de Fermat
2508:
2500:
2496:
2492:
2490:
2489:
2484:
2482:
2481:
2465:
2451:
2447:
2437:
2427:
2425:
2424:
2419:
2417:
2416:
2398:
2397:
2381:
2377:
2375:
2374:
2369:
2367:
2366:
2348:
2347:
2331:
2329:
2328:
2323:
2318:
2317:
2312:
2306:
2305:
2287:
2286:
2281:
2275:
2274:
2262:
2251:may be written
2250:
2248:
2247:
2242:
2234:
2222:
2218:
2214:
2210:
2208:
2207:
2202:
2197:
2196:
2191:
2176:
2175:
2170:
2161:
2160:
2155:
2135:rational numbers
2120:
2079:
2065:
2061:
2051:
2048:, one says that
2047:
2043:
2037:
2033:
2029:
2025:
2021:
2017:
2005:
2001:
1980:
1976:
1972:
1968:
1964:
1956:
1952:
1948:
1929:
1925:
1921:
1917:
1909:
1905:
1901:
1885:
1883:
1882:
1877:
1875:
1874:
1856:
1855:
1839:
1837:
1836:
1831:
1820:
1819:
1814:
1799:
1798:
1793:
1780:
1778:
1777:
1772:
1764:
1763:
1745:
1744:
1728:
1726:
1725:
1720:
1715:
1714:
1709:
1703:
1702:
1684:
1683:
1678:
1672:
1671:
1659:
1658:
1653:
1647:
1646:
1630:
1626:
1622:
1618:
1614:
1600:
1576:
1550:
1540:
1534:
1505:
1503:
1502:
1497:
1492:
1484:
1472:
1470:
1469:
1464:
1446:
1444:
1443:
1438:
1436:
1428:
1412:
1410:
1409:
1404:
1399:
1388:
1363:
1361:
1360:
1355:
1350:
1342:
1326:
1324:
1323:
1318:
1313:
1305:
1287:
1285:
1284:
1279:
1268:
1256:
1254:
1253:
1248:
1228:
1226:
1225:
1220:
1212:
1198:
1190:
1182:
1163:
1145:
1136:
1134:
1129:
1104:
1069:
1031:
1023:
1019:
1002:
974:
960:
952:additive inverse
948:
937:
923:Inverse elements
916:
906:
886:
872:Identity element
866:
839:
798:
794:
790:
786:
782:
778:
772:
766:
752:
743:
739:
735:
729:
725:
706:
696:
692:
688:
682:
660:
652:
639:binary operation
637:together with a
636:
628:
604:
597:
590:
379:Commutative ring
308:Rack and quandle
273:
272:
240:field extensions
238:, which include
227:as a dimension.
213:polynomial rings
73:
59:
53:
21:
20090:
20089:
20085:
20084:
20083:
20081:
20080:
20079:
20045:
20044:
20041:
20036:
20018:
19980:
19936:
19873:
19825:
19767:
19758:
19724:Change of basis
19714:Multilinear map
19652:
19634:
19629:
19582:
19579:
19574:
19552:
19529:
19527:
19523:
19500:10.1.1.318.4292
19482:
19472:
19450:
19412:Spivak, Michael
19354:
19332:
19301:
19291:Springer-Verlag
19274:
19252:
19226:
19208:Kreyszig, Erwin
19201:
19191:Springer-Verlag
19179:
19169:Springer-Verlag
19157:
19126:10.2307/2035388
19103:
19093:Springer-Verlag
19085:Halmos, Paul R.
19078:
19053:
19023:
19013:Springer-Verlag
19005:Eisenbud, David
18990:10.2307/2320587
18968:
18958:Springer-Verlag
18943:
18933:Springer-Verlag
18918:
18908:Springer-Verlag
18885:
18866:
18829:
18804:
18776:
18771:
18742:
18741:
18733:Peano, Giuseppe
18670:
18650:
18553:, Verona: 53–61
18495:
18483:
18478:
18449:
18424:
18397:
18371:
18343:
18333:Springer-Verlag
18318:
18285:
18267:Kreyszig, Erwin
18260:
18246:Kreyszig, Erwin
18239:
18221:
18203:
18185:
18177:, Brooks-Cole,
18167:
18145:
18123:
18105:
18062:"Tangent plane"
18060:BSE-3 (2001) ,
18054:
18044:Springer-Verlag
18032:
18022:Springer-Verlag
18007:
17997:Springer-Verlag
17983:
17978:
17973:
17963:Springer-Verlag
17939:
17919:
17901:
17891:Springer-Verlag
17865:
17841:
17812:
17787:
17761:
17732:
17712:
17681:Halmos, Paul R.
17674:
17648:
17626:
17593:
17588:
17583:
17575:
17571:
17563:
17559:
17551:
17547:
17539:
17535:
17527:
17523:
17515:
17511:
17503:
17499:
17491:
17487:
17479:
17475:
17467:
17463:
17455:
17451:
17443:
17439:
17431:
17427:
17419:
17415:
17407:
17403:
17395:
17391:
17383:
17379:
17371:
17367:
17359:
17355:
17347:
17343:
17335:
17331:
17323:
17319:
17311:
17307:
17299:
17295:
17287:
17283:
17275:
17271:
17263:
17259:
17251:
17247:
17239:
17235:
17227:
17223:
17215:
17211:
17199:
17195:
17187:
17183:
17175:
17171:
17167:, ch. 2, p. 48.
17163:
17159:
17151:
17147:
17139:
17135:
17123:
17119:
17111:
17107:
17099:
17095:
17087:
17083:
17075:
17071:
17067:, ch. 2, p. 48.
17063:
17059:
17051:
17047:
17043:, ch. 3, p. 64.
17039:
17035:
17027:
17023:
17019:, ch. 1, p. 35.
17015:
17011:
17007:, ch. 1, p. 29.
17003:
16999:
16991:
16987:
16981:& Lang 1987
16979:
16975:
16967:
16963:
16955:
16951:
16943:
16939:
16931:
16927:
16919:
16915:
16907:
16903:
16899:, p. 28, Ex. 9.
16895:
16891:
16883:
16879:
16871:
16867:
16859:
16855:
16847:
16843:
16839:, ch. 2, p. 45.
16835:
16831:
16823:
16819:
16811:
16807:
16799:
16795:
16787:
16783:
16771:
16767:
16755:
16751:
16739:
16735:
16723:
16719:
16711:
16707:
16699:
16695:
16687:
16683:
16675:
16671:
16663:
16659:
16651:
16647:
16639:
16635:
16629:Bellavitis 1833
16627:
16623:
16615:
16611:
16603:
16599:
16591:
16587:
16575:
16571:
16559:
16555:
16543:
16539:
16531:
16527:
16511:
16507:
16499:
16495:
16483:
16476:
16468:
16464:
16456:
16452:
16444:
16440:
16432:
16428:
16420:
16416:
16400:
16396:
16388:
16384:
16380:, ch. 1, p. 27.
16376:
16372:
16364:
16360:
16352:
16348:
16344:
16339:
16318:
16314:
16301:
16291:
16287:
16278:
16274:
16245:
16241:
16239:
16236:
16235:
16211:
16208:
16207:
16205:
16201:
16197:, §5.3, p. 125.
16177:
16173:
16171:
16168:
16167:
16150:
16146:
16144:
16141:
16140:
16138:
16134:
16102:
16098:
16083:
16079:
16064:
16060:
16046:
16043:
16042:
16036:
16032:
16028:, loc = ch. II.
16026:Bourbaki (1989)
16019:
16015:
15993:
15988:
15985:
15984:
15974:
15970:
15957:
15953:
15944:
15940:
15923:
15919:
15891:
15890:
15888:
15885:
15884:
15882:
15878:
15874:
15812:
15796:
15788:
15786:
15783:
15782:
15762:
15754:
15749:
15746:
15745:
15734:
15724:
15702:
15692:
15661:
15653:
15642:
15634:
15610:
15607:
15606:
15599:free transitive
15571:
15563:Main articles:
15561:
15484:
15478:
15462:cotangent space
15394:
15365:
15340:
15322:
15269:
15255:
15240:
15196:
15193:
15192:
15159:
15149:
15141:Main articles:
15139:
15134:
15108:
15103:
15102:
15093:
15088:
15087:
15075:
15070:
15069:
15060:
15055:
15054:
15052:
15049:
15048:
15024:
15019:
15018:
15009:
15004:
15003:
15001:
14998:
14997:
14980:
14975:
14974:
14965:
14960:
14959:
14957:
14954:
14953:
14950:tensor products
14929:
14926:
14925:
14902:
14897:
14896:
14881:
14876:
14875:
14866:
14861:
14860:
14858:
14855:
14854:
14835:
14832:
14831:
14803:
14800:
14799:
14769:
14764:
14763:
14761:
14758:
14757:
14704:
14701:
14700:
14699:matrices, with
14684:
14681:
14680:
14664:
14661:
14660:
14653:Jacobi identity
14546:
14543:
14542:
14487:
14484:
14483:
14464:
14461:
14460:
14444:
14441:
14440:
14412:
14409:
14408:
14384:polynomial ring
14358:
14355:
14354:
14347:is specified).
14310:
14308:
14305:
14304:
14264:
14244:
14242:
14239:
14238:
14235:coordinate ring
14206:
14203:
14202:
14191:
14183:Main articles:
14181:
14127:Euclidean space
14095:
14092:
14091:
14062:
14059:
14058:
14022:
14019:
14018:
13998:
13995:
13994:
13970:
13966:
13964:
13961:
13960:
13956:is a key case.
13929:
13926:
13925:
13891:
13889:
13887:
13884:
13883:
13843:
13841:
13823:
13819:
13793:
13790:
13789:
13760:
13756:
13754:
13751:
13750:
13731:
13725:
13676:
13669:
13649:
13645:
13629:
13625:
13624:
13618:
13614:
13602:
13596:
13593:
13592:
13564:
13560:
13558:
13555:
13554:
13529:
13526:
13525:
13491:
13484:
13464:
13460:
13442:
13438:
13437:
13433:
13432:
13426:
13422:
13401:
13395:
13392:
13391:
13365:
13361:
13355:
13351:
13346:
13343:
13342:
13319:
13315:
13300:
13296:
13287:
13283:
13281:
13278:
13277:
13236:
13232:
13230:
13227:
13226:
13223:Lebesgue spaces
13196:
13192:
13184:
13181:
13180:
13164:
13161:
13160:
13126:
13104:
13097:
13092:
13091:
13074:
13068:
13064:
13063:
13059:
13058:
13049:
13045:
13037:
13034:
13033:
13014:
13000:
12997:
12996:
12968:
12964:
12955:
12951:
12939:
12935:
12927:
12923:
12922:
12911:
12898:
12894:
12888:
12883:
12882:
12877:
12874:
12873:
12844:
12840:
12819:
12815:
12800:
12796:
12790:
12785:
12784:
12779:
12776:
12775:
12751:
12748:
12747:
12722:
12719:
12718:
12695:
12692:
12691:
12671:
12667:
12665:
12662:
12661:
12660:components are
12644:
12640:
12638:
12635:
12634:
12588:
12584:
12566:
12562:
12550:
12546:
12545:
12541:
12532:
12527:
12526:
12524:
12521:
12520:
12501:
12498:
12497:
12480:
12476:
12474:
12471:
12470:
12442: for
12440:
12428:
12417:
12412:
12411:
12405:
12401:
12396:
12390:
12385:
12381:
12380:
12371:
12367:
12362:
12357:
12354:
12353:
12340: and
12338:
12323: for
12321:
12315:
12309:
12305:
12300:
12294:
12281:
12277:
12272:
12267:
12264:
12263:
12229:
12226:
12225:
12208:
12205:
12204:
12175:
12171:
12156:
12152:
12143:
12139:
12138:
12134:
12126:
12124:
12121:
12120:
12102:
12098:
12096:
12093:
12092:
12075:
12069:
12043:
12041:
12038:
12037:
12021:
12013:
12010:
12009:
11992:
11988:
11986:
11983:
11982:
11955:
11952:
11951:
11923:
11918:
11917:
11915:
11912:
11911:
11895:
11892:
11891:
11875:
11872:
11871:
11845:
11840:
11831:
11826:
11825:
11820:
11808:
11802:
11799:
11798:
11797:if and only if
11782:
11780:
11777:
11776:
11759:
11754:
11753:
11751:
11748:
11747:
11719:
11716:
11715:
11679:
11676:
11675:
11640:
11637:
11636:
11629:Cauchy sequence
11605:
11602:
11601:
11570:
11567:
11566:
11548:
11545:
11544:
11526:
11521:
11520:
11518:
11515:
11514:
11489:function series
11469:
11466:
11465:
11445:
11441:
11439:
11436:
11435:
11416:
11413:
11412:
11411:of elements of
11389:
11385:
11376:
11372:
11370:
11367:
11366:
11345:
11341:
11326:
11322:
11310:
11291:
11287:
11281:
11270:
11264:
11261:
11260:
11230:
11227:
11226:
11207:
11202:
11199:
11198:
11182:
11174:
11172:
11169:
11168:
11152:
11149:
11148:
11132:
11129:
11128:
11112:
11109:
11108:
11092:
11090:
11087:
11086:
11070:
11068:
11065:
11064:
11061:continuous maps
11036:
11033:
11032:
11029:
11023:
10999:
10995:
10989:
10985:
10976:
10972:
10966:
10962:
10953:
10949:
10943:
10939:
10930:
10926:
10920:
10916:
10902:
10897:
10892:
10887:
10884:
10883:
10826:
10824:
10821:
10820:
10801:
10796:
10791:
10786:
10783:
10782:
10758:
10754:
10748:
10744:
10735:
10731:
10725:
10721:
10712:
10708:
10702:
10698:
10689:
10685:
10679:
10675:
10664:
10659:
10654:
10649:
10646:
10645:
10628:
10623:
10622:
10620:
10617:
10616:
10613:Minkowski space
10583:
10575:
10570:
10567:
10566:
10547:
10542:
10537:
10529:
10524:
10519:
10503:
10495:
10488:
10484:
10470:
10462:
10460:
10457:
10456:
10433:
10431:
10428:
10427:
10411:
10409:
10406:
10405:
10385:
10380:
10379:
10377:
10374:
10373:
10353:
10349:
10343:
10339:
10324:
10320:
10314:
10310:
10302:
10294:
10283:
10275:
10270:
10267:
10266:
10245:
10241:
10239:
10236:
10235:
10199:
10191:
10186:
10178:
10173:
10168:
10166:
10163:
10162:
10161:
10138:
10130:
10125:
10122:
10121:
10120:
10104:
10099:
10094:
10092:
10089:
10088:
10077:
10069:Main articles:
10067:
10045:
10041:
10039:
10036:
10035:
10019:
10016:
10015:
9998:
9994:
9992:
9989:
9988:
9968:
9964:
9955:
9951:
9943:
9940:
9939:
9910:
9905:
9904:
9902:
9899:
9898:
9882:
9879:
9878:
9857:
9854:
9853:
9837:infinite series
9824:
9793:
9785:
9768:
9760:
9752:
9749:
9748:
9730:
9727:
9726:
9710:
9707:
9706:
9683:
9680:
9679:
9642:
9639:
9638:
9618:
9615:
9614:
9594:
9586:
9584:
9581:
9580:
9561:
9553:
9548:
9545:
9544:
9519:
9516:
9515:
9493:
9490:
9489:
9473:
9470:
9469:
9455:
9452:
9443:
9438:
9437:
9429:
9420:
9415:
9414:
9406:
9396:
9384:
9379:
9378:
9369:
9364:
9363:
9352:
9349:
9346:
9341:
9332:
9327:
9326:
9318:
9309:
9304:
9303:
9293:
9285:
9273:
9268:
9267:
9258:
9253:
9252:
9246:
9245:
9240:
9232:
9223:
9212:
9195:
9181:
9170:
9151:
9140:
9132:
9119:
9117:
9114:
9113:
9093:
9088:
9087:
9078:
9073:
9072:
9057:
9052:
9051:
9042:
9037:
9036:
9027:
9022:
9021:
9012:
9007:
9006:
9004:
9001:
9000:
8978:
8975:
8974:
8938:
8936:
8933:
8932:
8913:
8905:
8891:
8889:
8886:
8885:
8869:
8867:
8864:
8863:
8844:
8842:
8839:
8838:
8822:
8820:
8817:
8816:
8800:
8797:
8796:
8770:
8767:
8766:
8729:
8726:
8725:
8705:
8702:
8701:
8685:
8682:
8681:
8656:
8653:
8652:
8629:
8625:
8620:
8617:
8616:
8610:
8604:
8583:
8580:
8579:
8562:
8558:
8546:
8540:
8537:
8536:
8515:
8511:
8499:
8493:
8490:
8489:
8469:
8465:
8463:
8460:
8459:
8442:
8437:
8436:
8434:
8431:
8430:
8414:
8411:
8410:
8391:
8388:
8387:
8364:
8354:
8349:
8348:
8344:
8343:
8341:
8338:
8337:
8320:
8316:
8314:
8311:
8310:
8291:
8287:
8275:
8270:
8267:
8264:
8263:
8246:
8238:Main articles:
8236:
8176:
8171:
8168:
8167:
8131:
8128:
8127:
8114:
8108:
8077:
8074:
8073:
8053:
8050:
8049:
8048:for a constant
8032:
8028:
8013:
8009:
7995:
7992:
7991:
7974:
7970:
7961:
7957:
7948:
7944:
7930:
7927:
7926:
7909:
7905:
7890:
7886:
7884:
7881:
7880:
7864:
7861:
7860:
7833:
7829:
7825:
7815:
7811:
7810:
7808:
7802:
7798:
7792:
7781:
7754:
7751:
7750:
7731:
7728:
7727:
7710:
7706:
7704:
7701:
7700:
7671:
7667:
7663:
7653:
7649:
7648:
7646:
7640:
7636:
7618:
7614:
7610:
7600:
7596:
7595:
7593:
7587:
7583:
7570:
7562:
7560:
7554:
7550:
7538:
7534:
7532:
7529:
7528:
7512:
7509:
7508:
7492:
7484:
7479:
7476:
7475:
7459:
7457:
7454:
7453:
7437:
7434:
7433:
7417:
7406:
7404:
7401:
7400:
7381:
7378:
7377:
7361:
7358:
7357:
7332:
7321:
7292:
7289:
7288:
7269:
7266:
7265:
7249:
7247:
7244:
7243:
7227:
7225:
7222:
7221:
7193:
7190:
7189:
7161:
7158:
7157:
7135:
7132:
7131:
7115:
7112:
7111:
7094:
7089:
7088:
7086:
7083:
7082:
7065:
7060:
7059:
7057:
7054:
7053:
7027:
7022:
7021:
7006:
7001:
7000:
6998:
6995:
6994:
6969:
6943:
6932:
6929:
6928:
6900:
6895:
6894:
6885:
6880:
6879:
6878:
6874:
6872:
6869:
6868:
6845:
6840:
6839:
6837:
6834:
6833:
6810:
6805:
6804:
6802:
6799:
6798:
6782:
6779:
6778:
6762:
6760:
6757:
6756:
6728:
6720:
6712:
6695:
6693:
6690:
6689:
6673:
6670:
6669:
6651:
6648:
6647:
6628:
6623:
6620:
6619:
6597:
6594:
6593:
6558:
6555:
6554:
6515:
6512:
6511:
6510:of elements of
6491:
6488:
6487:
6471:
6468:
6467:
6447:
6444:
6443:
6427:
6424:
6423:
6407:
6404:
6403:
6383:
6380:
6379:
6376:linear subspace
6359:
6356:
6355:
6339:
6337:
6334:
6333:
6317:
6314:
6313:
6297:
6294:
6293:
6274:
6263:
6257:Linear subspace
6255:Main articles:
6253:
6245:
6239:) in question.
6234:
6224:
6210:
6200:
6191:
6185:
6184:. If the field
6179:
6169:
6126:
6123:
6122:
6116:
6110:
6104:
6093:
6079:
6069:
6063:
6054:
6044:
6026:
6020:
6009:
6003:
5997:
5983:
5977:
5971:
5948:
5942:
5931:
5921:
5915:
5912:
5906:
5903:
5897:
5872:
5862:
5856:
5853:
5835:
5824:
5822:
5819:
5818:
5800:
5798:
5795:
5794:
5778:
5775:
5774:
5750:
5747:
5746:
5721:
5717:
5708:
5704:
5698:
5687:
5668:
5664:
5655:
5651:
5645:
5634:
5621:
5617:
5608:
5604:
5598:
5587:
5582:
5578:
5566:
5562:
5547:
5543:
5534:
5530:
5519:
5517:
5514:
5513:
5507:
5501:
5485:
5482:
5481:
5475:
5469:
5455:
5447:Main articles:
5445:
5434:
5428:
5422:
5416:
5402:
5396:
5390:
5384:
5366:
5360:
5354:
5348:
5333:
5327:
5313:
5307:
5290:
5280:
5268:
5262:
5250:
5238:
5237:, also denoted
5226:
5220:
5210:
5201:
5195:
5189:
5183:
5177:
5165:
5159:
5153:
5139:
5136:§ Examples
5125:
5119:
5113:
5100:
5094:
5088:
5082:
5072:
5066:
5052:
5030:
5012:
4995:
4978:
4951:
4948:
4947:
4931:
4928:
4927:
4908:
4905:
4904:
4888:
4886:
4883:
4882:
4866:
4864:
4861:
4860:
4846:
4845:
4837:
4818:
4810:
4795:
4794:
4783:
4766:
4753:
4745:
4737:
4727:
4725:
4722:
4721:
4706:
4700:
4674:
4670:
4668:
4665:
4664:
4648:
4645:
4644:
4628:
4625:
4624:
4601:
4597:
4579:
4575:
4555:
4552:
4551:
4548:
4502:
4498:
4474:
4470:
4468:
4465:
4464:
4424:
4422:
4419:
4418:
4398:
4393:
4390:
4389:
4353:
4350:
4349:
4333:
4331:
4328:
4327:
4309:
4308:
4303:
4298:
4292:
4291:
4286:
4281:
4271:
4270:
4262:
4259:
4258:
4255:
4237:
4229:
4224:
4221:
4220:
4195:
4178:
4175:
4174:
4152:
4141:
4138:
4137:
4119:
4116:
4115:
4101:
4100:
4091:
4086:
4080:
4072:
4058:
4051:
4045:
4044:
4035:
4030:
4027:
4019:
4005:
3995:
3993:
3990:
3989:
3982:
3972:Linear equation
3970:Main articles:
3968:
3940:
3924:
3854:
3851:
3850:
3822:
3819:
3818:
3812:
3806:
3800:
3796:
3719:
3716:
3715:
3699:
3691:
3677:
3674:
3673:
3666:
3660:
3658:Function spaces
3649:
3628:
3617:
3615:
3612:
3611:
3605:
3599:
3593:
3587:
3578:
3551:
3538:
3528:
3522:
3516:
3510:
3504:
3469:
3429:
3419:
3413:
3407:
3394:
3388:
3386:complex numbers
3382:
3366:
3356:
3350:
3343:
3333:
3327:
3324:
3316:
3296:
3292:
3277:
3273:
3264:
3260:
3255:
3252:
3251:
3245:
3237:
3230:
3224:
3221:
3196:
3195:
3164:
3143:
3142:
3130:
3126:
3117:
3113:
3104:
3100:
3091:
3087:
3077:
3068:
3064:
3055:
3051:
3036:
3032:
3023:
3019:
3012:
3010:
3007:
3006:
2994:
2986:
2982:
2978:
2974:
2971:
2960:
2949:
2939:
2932:
2926:
2920:
2911:
2904:
2892:
2886:
2877:
2871:
2865:
2855:
2849:
2843:
2830:
2816:
2810:
2791:
2790:
2789:
2788:
2787:
2780:
2773:
2770:
2761:
2760:
2759:
2758:(red) is shown.
2753:
2747:
2737:
2734:
2723:
2718:
2712:
2672:function spaces
2652:scalar products
2632:matrix notation
2630:introduced the
2589:complex numbers
2530:affine geometry
2526:
2506:
2498:
2494:
2477:
2473:
2471:
2468:
2467:
2461:
2449:
2443:
2433:
2428:are called the
2412:
2408:
2393:
2389:
2387:
2384:
2383:
2379:
2362:
2358:
2343:
2339:
2337:
2334:
2333:
2313:
2308:
2307:
2301:
2297:
2282:
2277:
2276:
2270:
2266:
2258:
2256:
2253:
2252:
2230:
2228:
2225:
2224:
2220:
2216:
2212:
2192:
2187:
2186:
2171:
2166:
2165:
2156:
2151:
2150:
2145:
2142:
2141:
2127:axiom of choice
2077:
2063:
2059:
2049:
2045:
2044:is the span of
2041:
2039:
2035:
2031:
2027:
2023:
2019:
2015:
2003:
1999:
1998:Given a subset
1982:
1978:
1974:
1970:
1966:
1962:
1954:
1950:
1946:
1944:vector subspace
1940:linear subspace
1934:Linear subspace
1927:
1923:
1919:
1915:
1910:are said to be
1907:
1903:
1899:
1886:are called the
1870:
1866:
1851:
1847:
1845:
1842:
1841:
1815:
1810:
1809:
1794:
1789:
1788:
1786:
1783:
1782:
1759:
1755:
1740:
1736:
1734:
1731:
1730:
1710:
1705:
1704:
1698:
1694:
1679:
1674:
1673:
1667:
1663:
1654:
1649:
1648:
1642:
1638:
1636:
1633:
1632:
1628:
1624:
1620:
1616:
1612:
1599:
1592:
1582:
1575:
1565:
1552:
1546:
1536:
1530:
1523:
1488:
1480:
1478:
1475:
1474:
1452:
1449:
1448:
1432:
1424:
1419:
1416:
1415:
1395:
1384:
1370:
1367:
1366:
1346:
1338:
1333:
1330:
1329:
1309:
1301:
1296:
1293:
1292:
1264:
1262:
1259:
1258:
1236:
1233:
1232:
1208:
1194:
1186:
1178:
1176:
1173:
1172:
1168:of this group.
1159:
1158:from the field
1143:
1132:
1131:
1127:
1120:complex numbers
1077:
1043:
1029:
1021:
1010:
983:
962:
956:
939:
929:
908:
894:
878:
850:
815:
792:
788:
784:
780:
774:
768:
762:
745:
741:
737:
731:
727:
723:
698:
694:
690:
684:
678:
671:vector addition
658:
650:
643:binary function
634:
629:is a non-empty
626:
617:
608:
579:
578:
577:
548:Non-associative
530:
519:
518:
508:
488:
477:
476:
465:Map of lattices
461:
457:Boolean algebra
452:Heyting algebra
426:
415:
414:
408:
389:Integral domain
353:
342:
341:
335:
289:
221:function spaces
135:complex numbers
90:(also called a
65:
55:
49:
42:
35:
28:
23:
22:
15:
12:
11:
5:
20088:
20078:
20077:
20072:
20067:
20062:
20057:
20038:
20037:
20035:
20034:
20023:
20020:
20019:
20017:
20016:
20011:
20006:
20001:
19996:
19994:Floating-point
19990:
19988:
19982:
19981:
19979:
19978:
19976:Tensor product
19973:
19968:
19963:
19961:Function space
19958:
19953:
19947:
19945:
19938:
19937:
19935:
19934:
19929:
19924:
19919:
19914:
19909:
19904:
19899:
19897:Triple product
19894:
19889:
19883:
19881:
19875:
19874:
19872:
19871:
19866:
19861:
19856:
19851:
19846:
19841:
19835:
19833:
19827:
19826:
19824:
19823:
19818:
19813:
19811:Transformation
19808:
19803:
19801:Multiplication
19798:
19793:
19788:
19783:
19777:
19775:
19769:
19768:
19761:
19759:
19757:
19756:
19751:
19746:
19741:
19736:
19731:
19726:
19721:
19716:
19711:
19706:
19701:
19696:
19691:
19686:
19681:
19676:
19671:
19666:
19660:
19658:
19657:Basic concepts
19654:
19653:
19651:
19650:
19645:
19639:
19636:
19635:
19632:Linear algebra
19628:
19627:
19620:
19613:
19605:
19599:
19598:
19584:"Vector space"
19578:
19577:External links
19575:
19573:
19572:
19550:
19534:
19475:
19470:
19453:
19448:
19420:
19408:
19365:
19352:
19335:
19330:
19304:
19299:
19277:
19272:
19255:
19250:
19229:
19224:
19204:
19199:
19182:
19177:
19160:
19155:
19145:(6 ed.),
19138:
19119:(3): 670–673,
19106:
19101:
19081:
19076:
19056:
19051:
19034:
19021:
19001:
18984:(7): 572–574,
18971:
18966:
18946:
18941:
18921:
18916:
18896:
18883:
18859:
18856:Addison-Wesley
18840:
18827:
18819:Addison-Wesley
18807:
18802:
18782:Ashcroft, Neil
18777:
18775:
18772:
18770:
18769:
18755:
18729:
18718:(3): 262–303,
18705:
18687:
18673:
18668:
18653:
18648:
18618:
18604:
18585:(3): 227–261,
18568:
18556:
18539:
18525:
18489:Banach, Stefan
18484:
18482:
18479:
18477:
18476:
18473:Academic Press
18471:, Boston, MA:
18461:
18447:
18427:
18422:
18409:
18396:978-1584888666
18395:
18382:
18369:
18346:
18341:
18321:
18316:
18308:Addison-Wesley
18296:
18283:
18263:
18258:
18242:
18237:
18224:
18219:
18206:
18201:
18188:
18183:
18170:
18165:
18148:
18143:
18126:
18121:
18108:
18103:
18090:
18087:Academic Press
18085:, Boston, MA:
18075:
18057:
18052:
18035:
18030:
18010:
18005:
17984:
17982:
17979:
17977:
17976:
17971:
17951:
17947:Linear Algebra
17942:
17937:
17922:
17917:
17904:
17899:
17879:
17868:
17863:
17844:
17839:
17823:
17810:
17790:
17785:
17769:Linear algebra
17764:
17759:
17744:
17735:
17730:
17715:
17710:
17689:
17677:
17672:
17651:
17646:
17629:
17624:
17608:Artin, Michael
17604:
17594:
17592:
17589:
17587:
17584:
17582:
17581:
17569:
17557:
17545:
17533:
17521:
17509:
17497:
17495:, §34, p. 108.
17485:
17473:
17461:
17449:
17437:
17425:
17413:
17401:
17397:Griffiths 1995
17389:
17377:
17365:
17363:, Th. III.1.1.
17353:
17341:
17329:
17317:
17305:
17293:
17281:
17269:
17257:
17245:
17233:
17221:
17209:
17193:
17181:
17169:
17157:
17155:, pp. 204–205.
17145:
17133:
17117:
17105:
17093:
17081:
17077:Nicholson 2018
17069:
17057:
17045:
17033:
17029:Nicholson 2018
17021:
17009:
16997:
16985:
16973:
16961:
16949:
16937:
16925:
16913:
16901:
16889:
16877:
16865:
16861:Nicholson 2018
16853:
16841:
16829:
16817:
16805:
16793:
16781:
16765:
16749:
16733:
16717:
16705:
16693:
16681:
16669:
16665:Grassmann 2000
16657:
16645:
16633:
16621:
16609:
16597:
16585:
16569:
16553:
16537:
16525:
16505:
16493:
16474:
16462:
16450:
16438:
16426:
16414:
16394:
16382:
16370:
16358:
16345:
16343:
16340:
16338:
16337:
16312:
16285:
16272:
16259:
16256:
16253:
16248:
16244:
16224:
16221:
16218:
16215:
16199:
16180:
16176:
16153:
16149:
16132:
16105:
16101:
16097:
16094:
16091:
16086:
16082:
16078:
16075:
16072:
16067:
16063:
16059:
16056:
16053:
16050:
16030:
16013:
16000:
15996:
15992:
15968:
15951:
15938:
15926:scalar product
15917:
15904:
15898:
15895:
15875:
15873:
15870:
15858:flag manifolds
15799:
15795:
15791:
15765:
15761:
15757:
15753:
15668:
15664:
15660:
15656:
15652:
15649:
15645:
15641:
15637:
15633:
15629:
15626:
15623:
15620:
15617:
15614:
15560:
15557:
15480:Main article:
15477:
15474:
15387:tangent spaces
15383:tangent bundle
15212:
15209:
15206:
15203:
15200:
15147:Tangent bundle
15138:
15137:Vector bundles
15135:
15133:
15130:
15111:
15106:
15101:
15096:
15091:
15086:
15083:
15078:
15073:
15068:
15063:
15058:
15032:
15027:
15022:
15017:
15012:
15007:
14983:
14978:
14973:
14968:
14963:
14933:
14910:
14905:
14900:
14895:
14892:
14889:
14884:
14879:
14874:
14869:
14864:
14839:
14819:
14816:
14813:
14810:
14807:
14797:tensor algebra
14777:
14772:
14767:
14741:
14738:
14735:
14732:
14729:
14726:
14723:
14720:
14717:
14714:
14711:
14708:
14688:
14668:
14657:
14656:
14640:
14637:
14634:
14631:
14628:
14625:
14622:
14619:
14616:
14613:
14610:
14607:
14604:
14601:
14598:
14595:
14592:
14589:
14586:
14583:
14580:
14577:
14574:
14571:
14568:
14565:
14562:
14559:
14556:
14553:
14550:
14540:
14524:
14521:
14518:
14515:
14512:
14509:
14506:
14503:
14500:
14497:
14494:
14491:
14468:
14448:
14428:
14425:
14422:
14419:
14416:
14371:
14368:
14365:
14362:
14317:
14313:
14292:
14289:
14286:
14283:
14280:
14277:
14274:
14271:
14267:
14263:
14260:
14257:
14254:
14251:
14247:
14222:
14219:
14216:
14213:
14210:
14180:
14177:
14102:
14099:
14069:
14066:
14038:
14035:
14032:
14029:
14026:
14002:
13973:
13969:
13945:
13942:
13939:
13936:
13933:
13907:
13903:
13900:
13897:
13894:
13871:
13868:
13865:
13859:
13855:
13852:
13849:
13846:
13840:
13837:
13834:
13831:
13826:
13822:
13818:
13815:
13812:
13806:
13800:
13797:
13777:
13774:
13771:
13768:
13763:
13759:
13745:, in honor of
13743:Hilbert spaces
13727:Main article:
13724:
13723:Hilbert spaces
13721:
13717:Sobolev spaces
13698:
13695:
13691:
13688:
13685:
13682:
13679:
13672:
13667:
13663:
13660:
13657:
13652:
13648:
13644:
13641:
13638:
13635:
13632:
13628:
13621:
13617:
13611:
13608:
13605:
13601:
13580:
13577:
13574:
13569:
13563:
13542:
13539:
13536:
13533:
13513:
13510:
13506:
13503:
13500:
13497:
13494:
13487:
13482:
13478:
13475:
13472:
13467:
13463:
13459:
13456:
13453:
13450:
13445:
13441:
13436:
13429:
13425:
13419:
13416:
13413:
13407:
13404:
13400:
13379:
13376:
13373:
13368:
13364:
13358:
13354:
13350:
13330:
13327:
13322:
13318:
13314:
13311:
13308:
13303:
13299:
13295:
13290:
13286:
13275:
13255:
13252:
13249:
13246:
13241:
13235:
13210:
13207:
13204:
13199:
13195:
13191:
13188:
13168:
13139:
13133:
13130:
13124:
13119:
13116:
13113:
13110:
13107:
13100:
13095:
13090:
13087:
13084:
13081:
13077:
13071:
13067:
13062:
13057:
13052:
13048:
13044:
13041:
13017:
13013:
13010:
13007:
13004:
12982:
12979:
12974:
12971:
12967:
12963:
12958:
12954:
12950:
12945:
12942:
12938:
12930:
12926:
12920:
12917:
12914:
12910:
12906:
12901:
12897:
12891:
12886:
12881:
12861:
12858:
12855:
12850:
12847:
12843:
12839:
12836:
12833:
12830:
12825:
12822:
12818:
12814:
12811:
12808:
12803:
12799:
12793:
12788:
12783:
12764:
12761:
12758:
12755:
12735:
12732:
12729:
12726:
12702:
12699:
12677:
12674:
12670:
12647:
12643:
12622:
12618:
12614:
12611:
12608:
12605:
12602:
12599:
12594:
12591:
12587:
12583:
12580:
12577:
12572:
12569:
12565:
12561:
12556:
12553:
12549:
12544:
12540:
12535:
12530:
12508:
12505:
12483:
12479:
12456:
12453:
12450:
12447:
12435:
12432:
12426:
12420:
12415:
12408:
12404:
12399:
12393:
12389:
12384:
12379:
12374:
12370:
12365:
12361:
12337:
12334:
12331:
12328:
12318:
12312:
12308:
12303:
12297:
12293:
12289:
12284:
12280:
12275:
12271:
12251:
12248:
12245:
12242:
12239:
12236:
12233:
12212:
12190:
12186:
12183:
12178:
12174:
12170:
12167:
12164:
12159:
12155:
12151:
12146:
12142:
12137:
12133:
12129:
12105:
12101:
12071:Main article:
12068:
12065:
12046:
12024:
12020:
12017:
11995:
11991:
11968:
11965:
11962:
11959:
11931:
11926:
11921:
11899:
11879:
11855:
11852:
11848:
11843:
11839:
11834:
11829:
11823:
11817:
11814:
11811:
11807:
11785:
11762:
11757:
11735:
11732:
11729:
11726:
11723:
11695:
11692:
11689:
11686:
11683:
11659:
11656:
11653:
11650:
11647:
11644:
11612:
11609:
11589:
11586:
11583:
11580:
11577:
11574:
11552:
11529:
11524:
11511:Unit "spheres"
11476:
11473:
11463:function space
11448:
11444:
11423:
11420:
11400:
11397:
11392:
11388:
11384:
11379:
11375:
11348:
11344:
11340:
11337:
11334:
11329:
11325:
11319:
11316:
11313:
11309:
11302:
11294:
11290:
11284:
11279:
11276:
11273:
11269:
11234:
11214:
11210:
11206:
11185:
11181:
11177:
11156:
11136:
11116:
11095:
11073:
11063:. Roughly, if
11040:
11025:Main article:
11022:
11019:
11007:
11002:
10998:
10992:
10988:
10984:
10979:
10975:
10969:
10965:
10961:
10956:
10952:
10946:
10942:
10938:
10933:
10929:
10923:
10919:
10915:
10912:
10909:
10905:
10900:
10895:
10891:
10863:
10860:
10857:
10854:
10851:
10848:
10845:
10842:
10839:
10836:
10833:
10829:
10808:
10804:
10799:
10794:
10790:
10766:
10761:
10757:
10751:
10747:
10743:
10738:
10734:
10728:
10724:
10720:
10715:
10711:
10705:
10701:
10697:
10692:
10688:
10682:
10678:
10674:
10671:
10667:
10662:
10657:
10653:
10631:
10626:
10596:
10593:
10590:
10586:
10582:
10578:
10574:
10554:
10550:
10545:
10540:
10536:
10532:
10527:
10522:
10518:
10514:
10510:
10506:
10502:
10498:
10494:
10491:
10487:
10483:
10480:
10477:
10473:
10469:
10465:
10453:law of cosines
10440:
10436:
10414:
10393:
10388:
10383:
10361:
10356:
10352:
10346:
10342:
10338:
10335:
10332:
10327:
10323:
10317:
10313:
10309:
10305:
10301:
10297:
10293:
10290:
10286:
10282:
10278:
10274:
10248:
10244:
10211:
10206:
10202:
10198:
10194:
10190:
10185:
10181:
10176:
10171:
10148:
10145:
10141:
10137:
10133:
10129:
10107:
10102:
10097:
10066:
10063:
10048:
10044:
10023:
10001:
9997:
9976:
9971:
9967:
9963:
9958:
9954:
9950:
9947:
9930:, for example
9913:
9908:
9886:
9865:
9862:
9823:
9820:
9803:
9800:
9796:
9792:
9788:
9784:
9781:
9778:
9775:
9771:
9767:
9763:
9759:
9756:
9737:
9734:
9714:
9690:
9687:
9667:
9664:
9661:
9658:
9655:
9652:
9649:
9646:
9622:
9597:
9593:
9589:
9568:
9564:
9560:
9556:
9552:
9529:
9526:
9523:
9503:
9500:
9497:
9477:
9451:
9446:
9441:
9436:
9432:
9428:
9423:
9418:
9413:
9409:
9402:
9399:
9397:
9392:
9387:
9382:
9377:
9372:
9367:
9362:
9359:
9355:
9351:
9350:
9344:
9340:
9335:
9330:
9325:
9321:
9317:
9312:
9307:
9299:
9296:
9294:
9288:
9284:
9281:
9276:
9271:
9266:
9261:
9256:
9251:
9248:
9247:
9239:
9225:
9222:
9219:
9215:
9211:
9208:
9205:
9202:
9198:
9191:
9184:
9180:
9177:
9173:
9169:
9166:
9163:
9157:
9154:
9152:
9147:
9143:
9139:
9135:
9131:
9128:
9125:
9122:
9121:
9101:
9096:
9091:
9086:
9081:
9076:
9071:
9068:
9065:
9060:
9055:
9050:
9045:
9040:
9035:
9030:
9025:
9020:
9015:
9010:
8985:
8982:
8945:
8941:
8920:
8916:
8912:
8908:
8904:
8901:
8898:
8894:
8872:
8851:
8847:
8825:
8804:
8780:
8777:
8774:
8751:
8748:
8745:
8742:
8739:
8736:
8733:
8709:
8689:
8669:
8666:
8663:
8660:
8640:
8637:
8632:
8628:
8624:
8614:tensor product
8606:Main article:
8603:
8602:Tensor product
8600:
8587:
8565:
8561:
8555:
8552:
8549:
8545:
8518:
8514:
8508:
8505:
8502:
8498:
8472:
8468:
8445:
8440:
8418:
8395:
8373:
8370:
8367:
8362:
8357:
8352:
8347:
8323:
8319:
8294:
8290:
8284:
8281:
8278:
8274:
8261:direct product
8250:direct product
8240:Direct product
8235:
8232:
8215:
8212:
8209:
8206:
8203:
8199:
8195:
8192:
8189:
8186:
8183:
8179:
8175:
8135:
8096:
8093:
8090:
8087:
8084:
8081:
8057:
8035:
8031:
8027:
8024:
8021:
8016:
8012:
8008:
8005:
8002:
7999:
7977:
7973:
7969:
7964:
7960:
7956:
7951:
7947:
7943:
7940:
7937:
7934:
7912:
7908:
7904:
7901:
7896:
7893:
7889:
7868:
7844:
7836:
7832:
7828:
7823:
7818:
7814:
7805:
7801:
7795:
7790:
7787:
7784:
7780:
7776:
7773:
7770:
7767:
7764:
7761:
7758:
7738:
7735:
7713:
7709:
7688:
7685:
7682:
7674:
7670:
7666:
7661:
7656:
7652:
7643:
7639:
7635:
7632:
7629:
7621:
7617:
7613:
7608:
7603:
7599:
7590:
7586:
7582:
7576:
7573:
7568:
7565:
7557:
7553:
7549:
7546:
7541:
7537:
7516:
7495:
7491:
7487:
7483:
7462:
7441:
7420:
7416:
7413:
7409:
7385:
7365:
7345:
7342:
7339:
7335:
7331:
7328:
7324:
7320:
7317:
7314:
7311:
7308:
7305:
7302:
7299:
7296:
7273:
7252:
7230:
7209:
7206:
7203:
7200:
7197:
7177:
7174:
7171:
7168:
7165:
7139:
7119:
7097:
7092:
7068:
7063:
7050:if and only if
7038:
7035:
7030:
7025:
7020:
7017:
7014:
7009:
7004:
6982:
6979:
6976:
6972:
6968:
6965:
6962:
6959:
6956:
6953:
6950:
6946:
6942:
6939:
6936:
6916:
6913:
6909:
6903:
6898:
6893:
6888:
6883:
6877:
6856:
6853:
6848:
6843:
6821:
6818:
6813:
6808:
6786:
6765:
6744:
6741:
6738:
6735:
6731:
6727:
6723:
6719:
6715:
6711:
6708:
6705:
6702:
6698:
6677:
6655:
6635:
6631:
6627:
6607:
6604:
6601:
6568:
6565:
6562:
6519:
6495:
6475:
6451:
6431:
6411:
6398:, or simply a
6387:
6374:) is called a
6363:
6342:
6321:
6301:
6252:
6249:
6244:
6241:
6157:
6154:
6151:
6148:
6145:
6142:
6139:
6136:
6133:
6130:
5982:, linear maps
5973:Main article:
5970:
5967:
5919:
5910:
5901:
5894:parallelepiped
5842:
5838:
5834:
5831:
5827:
5803:
5782:
5773:of the matrix
5754:
5734:
5730:
5724:
5720:
5714:
5711:
5707:
5701:
5696:
5693:
5690:
5686:
5682:
5679:
5676:
5671:
5667:
5661:
5658:
5654:
5648:
5643:
5640:
5637:
5633:
5629:
5624:
5620:
5614:
5611:
5607:
5601:
5596:
5593:
5590:
5586:
5581:
5577:
5574:
5569:
5565:
5561:
5558:
5555:
5550:
5546:
5542:
5537:
5533:
5529:
5526:
5522:
5489:
5444:
5441:
5389:-vector space
5273:is called the
5222:
4958:
4955:
4935:
4915:
4912:
4891:
4869:
4844:
4840:
4836:
4833:
4830:
4827:
4824:
4821:
4819:
4817:
4813:
4809:
4806:
4803:
4800:
4797:
4796:
4793:
4790:
4786:
4782:
4779:
4776:
4773:
4769:
4765:
4762:
4759:
4756:
4754:
4752:
4748:
4744:
4740:
4736:
4733:
4730:
4729:
4702:Main article:
4699:
4696:
4677:
4673:
4652:
4632:
4612:
4607:
4604:
4600:
4596:
4593:
4590:
4585:
4582:
4578:
4574:
4571:
4568:
4565:
4562:
4559:
4537:
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4525:
4522:
4519:
4516:
4513:
4510:
4505:
4501:
4497:
4494:
4491:
4488:
4485:
4480:
4477:
4473:
4446:
4443:
4440:
4437:
4434:
4431:
4427:
4415:matrix product
4401:
4397:
4378:
4375:
4372:
4369:
4366:
4363:
4360:
4357:
4348:is the vector
4336:
4313:
4307:
4304:
4302:
4299:
4297:
4294:
4293:
4290:
4287:
4285:
4282:
4280:
4277:
4276:
4274:
4269:
4266:
4244:
4240:
4236:
4232:
4228:
4202:
4198:
4194:
4191:
4188:
4185:
4182:
4162:
4159:
4155:
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4148:
4145:
4126:
4123:
4099:
4096:
4092:
4090:
4087:
4085:
4081:
4079:
4075:
4073:
4070:
4067:
4063:
4060:
4056:
4053:
4050:
4047:
4046:
4043:
4040:
4036:
4034:
4031:
4028:
4026:
4022:
4020:
4017:
4014:
4010:
4007:
4003:
4000:
3997:
3967:
3964:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3879:
3876:
3873:
3870:
3867:
3864:
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3838:
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3832:
3829:
3826:
3780:
3777:
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3762:
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3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3702:
3698:
3694:
3690:
3687:
3684:
3681:
3664:Function space
3662:Main article:
3659:
3656:
3637:
3632:
3627:
3624:
3620:
3426:imaginary unit
3381:
3378:
3320:
3304:
3299:
3295:
3291:
3288:
3285:
3280:
3276:
3272:
3267:
3263:
3259:
3220:
3217:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3165:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3144:
3141:
3138:
3133:
3129:
3125:
3120:
3116:
3112:
3107:
3103:
3099:
3094:
3090:
3086:
3083:
3080:
3078:
3076:
3071:
3067:
3063:
3058:
3054:
3050:
3047:
3044:
3039:
3035:
3031:
3026:
3022:
3018:
3015:
3014:
2970:
2967:
2862:multiplication
2771:
2764:
2763:
2762:
2735:
2728:
2727:
2726:
2725:
2724:
2722:
2719:
2714:Main article:
2711:
2708:
2704:Hilbert spaces
2676:Henri Lebesgue
2568:introduced an
2538:René Descartes
2525:
2522:
2480:
2476:
2415:
2411:
2407:
2404:
2401:
2396:
2392:
2365:
2361:
2357:
2354:
2351:
2346:
2342:
2321:
2316:
2311:
2304:
2300:
2296:
2293:
2290:
2285:
2280:
2273:
2269:
2265:
2261:
2240:
2237:
2233:
2200:
2195:
2190:
2185:
2182:
2179:
2174:
2169:
2164:
2159:
2154:
2149:
2114:
2113:
2090:
2081:
2074:generating set
2030:. The span of
2022:that contains
2010:or simply the
1996:
1991:
1936:
1931:
1906:-vector space
1896:
1891:
1873:
1869:
1865:
1862:
1859:
1854:
1850:
1829:
1826:
1823:
1818:
1813:
1808:
1805:
1802:
1797:
1792:
1770:
1767:
1762:
1758:
1754:
1751:
1748:
1743:
1739:
1718:
1713:
1708:
1701:
1697:
1693:
1690:
1687:
1682:
1677:
1670:
1666:
1662:
1657:
1652:
1645:
1641:
1619:-vector space
1609:
1597:
1590:
1573:
1563:
1543:standard basis
1522:
1519:
1507:
1506:
1495:
1491:
1487:
1483:
1462:
1459:
1456:
1435:
1431:
1427:
1423:
1413:
1402:
1398:
1394:
1391:
1387:
1383:
1380:
1377:
1374:
1364:
1353:
1349:
1345:
1341:
1337:
1327:
1316:
1312:
1308:
1304:
1300:
1277:
1274:
1271:
1267:
1246:
1243:
1240:
1218:
1215:
1211:
1207:
1204:
1201:
1197:
1193:
1189:
1185:
1181:
1106:
1105:
1075:
1071:
1070:
1041:
1038:Distributivity
1034:
1033:
1008:
1004:
1003:
981:
977:
976:
926:
919:
918:
875:
868:
867:
848:
841:
840:
813:
806:
805:
802:
755:
754:
713:
712:
616:
613:
610:
609:
607:
606:
599:
592:
584:
581:
580:
576:
575:
570:
565:
560:
555:
550:
545:
539:
538:
537:
531:
525:
524:
521:
520:
517:
516:
513:Linear algebra
507:
506:
501:
496:
490:
489:
483:
482:
479:
478:
475:
474:
471:Lattice theory
467:
460:
459:
454:
449:
444:
439:
434:
428:
427:
421:
420:
417:
416:
407:
406:
401:
396:
391:
386:
381:
376:
371:
366:
361:
355:
354:
348:
347:
344:
343:
334:
333:
328:
323:
317:
316:
315:
310:
305:
296:
290:
284:
283:
280:
279:
264:Hilbert spaces
197:natural number
170:linear algebra
26:
9:
6:
4:
3:
2:
20087:
20076:
20075:Vector spaces
20073:
20071:
20068:
20066:
20063:
20061:
20058:
20056:
20053:
20052:
20050:
20043:
20033:
20025:
20024:
20021:
20015:
20012:
20010:
20009:Sparse matrix
20007:
20005:
20002:
20000:
19997:
19995:
19992:
19991:
19989:
19987:
19983:
19977:
19974:
19972:
19969:
19967:
19964:
19962:
19959:
19957:
19954:
19952:
19949:
19948:
19946:
19944:constructions
19943:
19939:
19933:
19932:Outermorphism
19930:
19928:
19925:
19923:
19920:
19918:
19915:
19913:
19910:
19908:
19905:
19903:
19900:
19898:
19895:
19893:
19892:Cross product
19890:
19888:
19885:
19884:
19882:
19880:
19876:
19870:
19867:
19865:
19862:
19860:
19859:Outer product
19857:
19855:
19852:
19850:
19847:
19845:
19842:
19840:
19839:Orthogonality
19837:
19836:
19834:
19832:
19828:
19822:
19819:
19817:
19816:Cramer's rule
19814:
19812:
19809:
19807:
19804:
19802:
19799:
19797:
19794:
19792:
19789:
19787:
19786:Decomposition
19784:
19782:
19779:
19778:
19776:
19774:
19770:
19765:
19755:
19752:
19750:
19747:
19745:
19742:
19740:
19737:
19735:
19732:
19730:
19727:
19725:
19722:
19720:
19717:
19715:
19712:
19710:
19707:
19705:
19702:
19700:
19697:
19695:
19692:
19690:
19687:
19685:
19682:
19680:
19677:
19675:
19672:
19670:
19667:
19665:
19662:
19661:
19659:
19655:
19649:
19646:
19644:
19641:
19640:
19637:
19633:
19626:
19621:
19619:
19614:
19612:
19607:
19606:
19603:
19595:
19591:
19590:
19585:
19581:
19580:
19569:
19565:
19561:
19557:
19553:
19547:
19543:
19539:
19535:
19526:on 2007-01-13
19522:
19518:
19514:
19510:
19506:
19501:
19496:
19492:
19488:
19481:
19476:
19473:
19467:
19463:
19462:Prentice Hall
19459:
19454:
19451:
19445:
19441:
19437:
19432:
19431:
19430:Galois Theory
19425:
19421:
19417:
19413:
19409:
19406:
19402:
19398:
19394:
19390:
19386:
19382:
19379:(in German),
19378:
19374:
19370:
19369:Schönhage, A.
19366:
19363:
19359:
19355:
19349:
19345:
19341:
19336:
19333:
19327:
19323:
19322:
19317:
19313:
19309:
19305:
19302:
19296:
19292:
19288:
19287:
19282:
19278:
19275:
19269:
19265:
19261:
19256:
19253:
19247:
19243:
19238:
19237:
19230:
19227:
19221:
19217:
19213:
19209:
19205:
19202:
19196:
19192:
19188:
19183:
19180:
19174:
19170:
19166:
19165:Fibre Bundles
19161:
19158:
19152:
19148:
19144:
19139:
19136:
19132:
19127:
19122:
19118:
19114:
19113:
19107:
19104:
19098:
19094:
19090:
19086:
19082:
19079:
19073:
19069:
19068:Prentice Hall
19065:
19061:
19057:
19054:
19048:
19044:
19040:
19035:
19032:
19028:
19024:
19018:
19014:
19010:
19006:
19002:
18999:
18995:
18991:
18987:
18983:
18979:
18978:
18972:
18969:
18963:
18959:
18955:
18951:
18947:
18944:
18938:
18934:
18930:
18926:
18922:
18919:
18913:
18909:
18905:
18901:
18897:
18894:
18890:
18886:
18880:
18876:
18872:
18865:
18860:
18857:
18853:
18849:
18845:
18841:
18838:
18834:
18830:
18824:
18820:
18816:
18812:
18808:
18805:
18799:
18794:
18793:
18787:
18783:
18779:
18778:
18768:
18764:
18760:
18756:
18752:
18746:
18738:
18734:
18730:
18726:
18721:
18717:
18713:
18712:
18706:
18703:on 2006-11-23
18702:
18698:
18697:
18692:
18688:
18684:
18683:
18678:
18674:
18671:
18665:
18661:
18660:
18654:
18651:
18645:
18641:
18637:
18629:
18628:
18623:
18619:
18615:
18614:
18609:
18605:
18602:
18598:
18593:
18588:
18584:
18580:
18579:
18574:
18569:
18565:
18561:
18557:
18552:
18548:
18544:
18540:
18536:
18535:
18530:
18526:
18523:
18519:
18515:
18511:
18507:
18504:(in French),
18503:
18502:
18494:
18490:
18486:
18485:
18474:
18470:
18466:
18462:
18458:
18454:
18450:
18444:
18440:
18436:
18432:
18428:
18425:
18419:
18415:
18410:
18406:
18402:
18398:
18392:
18388:
18383:
18380:
18376:
18372:
18366:
18361:
18356:
18352:
18347:
18344:
18338:
18334:
18330:
18326:
18322:
18319:
18313:
18309:
18305:
18304:Real analysis
18301:
18297:
18294:
18290:
18286:
18280:
18276:
18272:
18268:
18264:
18261:
18255:
18251:
18247:
18243:
18240:
18234:
18230:
18225:
18222:
18216:
18212:
18207:
18204:
18198:
18194:
18189:
18186:
18180:
18176:
18171:
18168:
18162:
18158:
18154:
18149:
18146:
18140:
18136:
18132:
18127:
18124:
18118:
18114:
18109:
18106:
18100:
18096:
18091:
18088:
18084:
18080:
18076:
18073:
18069:
18068:
18063:
18058:
18055:
18049:
18045:
18041:
18036:
18033:
18027:
18023:
18019:
18018:Integration I
18015:
18011:
18008:
18002:
17998:
17994:
17990:
17986:
17985:
17974:
17968:
17964:
17960:
17956:
17952:
17948:
17943:
17940:
17934:
17930:
17929:
17923:
17920:
17914:
17910:
17905:
17902:
17896:
17892:
17888:
17884:
17883:Roman, Steven
17880:
17876:
17875:
17869:
17866:
17860:
17856:
17852:
17851:
17845:
17842:
17836:
17832:
17828:
17824:
17821:
17817:
17813:
17807:
17803:
17799:
17795:
17791:
17788:
17782:
17778:
17774:
17770:
17765:
17762:
17756:
17752:
17751:
17745:
17741:
17736:
17733:
17727:
17724:, CRC Press,
17723:
17722:
17716:
17713:
17707:
17703:
17699:
17695:
17690:
17686:
17682:
17678:
17675:
17669:
17665:
17661:
17657:
17652:
17649:
17643:
17638:
17637:
17630:
17627:
17621:
17617:
17616:Prentice Hall
17613:
17609:
17605:
17601:
17596:
17595:
17578:
17573:
17566:
17561:
17554:
17549:
17542:
17537:
17530:
17525:
17518:
17513:
17506:
17501:
17494:
17493:Kreyszig 1991
17489:
17482:
17477:
17470:
17465:
17458:
17453:
17446:
17441:
17434:
17433:Eisenbud 1995
17429:
17422:
17417:
17411:, ch. XVII.3.
17410:
17405:
17398:
17393:
17387:, Chapter 11.
17386:
17385:Kreyszig 1999
17381:
17374:
17369:
17362:
17357:
17350:
17345:
17338:
17333:
17326:
17321:
17314:
17309:
17302:
17297:
17290:
17285:
17278:
17273:
17266:
17261:
17254:
17249:
17242:
17241:Kreyszig 1989
17237:
17230:
17229:Kreyszig 1989
17225:
17218:
17213:
17206:
17205:Bourbaki 1987
17202:
17197:
17190:
17185:
17178:
17173:
17166:
17165:Bourbaki 2004
17161:
17154:
17149:
17142:
17137:
17130:
17126:
17121:
17114:
17109:
17102:
17097:
17090:
17089:Mac Lane 1998
17085:
17078:
17073:
17066:
17061:
17054:
17049:
17042:
17037:
17030:
17025:
17018:
17013:
17006:
17001:
16994:
16989:
16982:
16977:
16970:
16965:
16958:
16953:
16946:
16941:
16934:
16929:
16922:
16917:
16910:
16905:
16898:
16893:
16886:
16881:
16874:
16869:
16862:
16857:
16850:
16845:
16838:
16833:
16826:
16821:
16814:
16809:
16802:
16797:
16790:
16785:
16778:
16774:
16769:
16762:
16761:358–359
16758:
16757:Kreyszig 2020
16753:
16746:
16742:
16741:Kreyszig 2020
16737:
16730:
16726:
16721:
16714:
16709:
16702:
16697:
16690:
16685:
16678:
16673:
16666:
16661:
16654:
16653:Hamilton 1853
16649:
16642:
16637:
16630:
16625:
16618:
16613:
16606:
16601:
16594:
16593:Bourbaki 1969
16589:
16582:
16578:
16573:
16566:
16562:
16557:
16550:
16546:
16541:
16534:
16529:
16522:
16518:
16514:
16509:
16502:
16497:
16490:
16486:
16481:
16479:
16472:, p. 92.
16471:
16466:
16459:
16454:
16448:, p. 94.
16447:
16442:
16435:
16434:Bourbaki 1998
16430:
16424:, p. 17.
16423:
16418:
16412:, p. 86.
16411:
16407:
16403:
16402:Springer 2000
16398:
16392:, p. 87.
16391:
16386:
16379:
16374:
16368:, p. 86.
16367:
16362:
16355:
16350:
16346:
16334:
16333:vector fields
16330:
16326:
16322:
16316:
16308:
16304:
16299:
16295:
16294:homeomorphism
16289:
16282:
16276:
16246:
16242:
16222:
16219:
16216:
16213:
16203:
16196:
16195:Dudley (1989)
16178:
16174:
16151:
16147:
16136:
16129:
16125:
16121:
16103:
16095:
16089:
16084:
16076:
16070:
16065:
16057:
16054:
16051:
16040:
16034:
16027:
16023:
16017:
15998:
15994:
15990:
15982:
15978:
15972:
15965:
15961:
15955:
15948:
15942:
15935:
15931:
15927:
15921:
15902:
15893:
15880:
15876:
15869:
15867:
15863:
15859:
15855:
15854:Grassmannians
15851:
15847:
15843:
15838:
15836:
15832:
15828:
15824:
15819:
15815:
15793:
15780:
15779:above section
15759:
15751:
15741:
15737:
15731:
15727:
15722:
15718:
15714:
15709:
15705:
15699:
15695:
15690:
15686:
15682:
15666:
15658:
15639:
15627:
15624:
15618:
15615:
15612:
15604:
15601:vector space
15600:
15596:
15595:affine spaces
15588:
15584:
15580:
15575:
15570:
15566:
15556:
15554:
15550:
15546:
15545:division ring
15542:
15538:
15534:
15530:
15526:
15522:
15521:
15517:
15512:
15511:abelian group
15508:
15504:
15500:
15496:
15492:
15488:
15483:
15473:
15471:
15467:
15463:
15459:
15454:
15452:
15449:
15445:
15442:
15438:
15434:
15430:
15425:
15421:
15418:
15414:
15410:
15406:
15401:
15397:
15392:
15388:
15384:
15379:
15377:
15372:
15368:
15364:
15360:
15356:
15352:
15347:
15343:
15338:
15333:
15329:
15325:
15320:
15316:
15312:
15309:
15306:, there is a
15305:
15301:
15297:
15293:
15290:a product of
15289:
15285:
15281:
15276:
15272:
15266:
15262:
15258:
15253:
15249:
15243:
15238:
15234:
15230:
15226:
15210:
15204:
15201:
15198:
15190:
15186:
15182:
15179:
15175:
15174:vector bundle
15166:
15162:
15158:
15153:
15148:
15144:
15143:Vector bundle
15129:
15127:
15109:
15099:
15094:
15084:
15081:
15076:
15066:
15061:
15046:
15030:
15025:
15015:
15010:
14981:
14971:
14966:
14951:
14947:
14931:
14924:
14908:
14903:
14893:
14890:
14887:
14882:
14872:
14867:
14853:
14837:
14814:
14808:
14798:
14793:
14791:
14790:cross product
14775:
14770:
14755:
14739:
14736:
14733:
14730:
14727:
14724:
14721:
14715:
14712:
14709:
14686:
14666:
14654:
14638:
14635:
14626:
14623:
14620:
14614:
14611:
14605:
14596:
14593:
14590:
14584:
14581:
14575:
14566:
14563:
14560:
14554:
14551:
14541:
14538:
14519:
14516:
14513:
14507:
14504:
14498:
14495:
14492:
14482:
14481:
14480:
14466:
14446:
14423:
14420:
14417:
14406:
14405:
14399:
14397:
14393:
14389:
14385:
14366:
14360:
14353:
14348:
14346:
14342:
14338:
14334:
14315:
14290:
14284:
14281:
14278:
14275:
14272:
14265:
14258:
14255:
14252:
14236:
14220:
14217:
14214:
14211:
14208:
14200:
14195:
14190:
14186:
14176:
14173:
14169:
14164:
14160:
14156:
14152:
14151:wavefunctions
14148:
14144:
14140:
14135:
14130:
14128:
14124:
14120:
14116:
14100:
14097:
14089:
14085:
14084:
14067:
14064:
14056:
14052:
14033:
14030:
14027:
14016:
14000:
13993:
13989:
13971:
13967:
13957:
13943:
13937:
13931:
13923:
13898:
13892:
13869:
13866:
13863:
13850:
13844:
13835:
13829:
13820:
13816:
13810:
13804:
13798:
13775:
13761:
13757:
13748:
13747:David Hilbert
13744:
13735:
13730:
13729:Hilbert space
13720:
13718:
13714:
13709:
13696:
13693:
13686:
13680:
13677:
13670:
13665:
13658:
13650:
13646:
13642:
13636:
13630:
13626:
13615:
13603:
13567:
13561:
13537:
13531:
13511:
13508:
13501:
13495:
13492:
13485:
13480:
13473:
13465:
13461:
13457:
13451:
13443:
13439:
13434:
13423:
13411:
13405:
13402:
13377:
13371:
13366:
13356:
13352:
13328:
13325:
13320:
13316:
13312:
13309:
13306:
13301:
13297:
13293:
13288:
13284:
13273:
13271:
13266:
13253:
13239:
13233:
13224:
13208:
13202:
13197:
13189:
13159:
13155:
13152:The space of
13150:
13137:
13131:
13128:
13122:
13114:
13108:
13105:
13098:
13085:
13079:
13065:
13060:
13055:
13050:
13042:
13032:
13005:
13002:
12993:
12980:
12977:
12972:
12969:
12965:
12961:
12956:
12952:
12948:
12943:
12940:
12936:
12928:
12924:
12918:
12915:
12912:
12908:
12904:
12899:
12889:
12859:
12856:
12848:
12845:
12841:
12837:
12831:
12828:
12823:
12820:
12816:
12806:
12791:
12762:
12759:
12756:
12753:
12733:
12727:
12724:
12716:
12700:
12697:
12675:
12672:
12668:
12645:
12641:
12620:
12616:
12612:
12609:
12606:
12603:
12600:
12597:
12592:
12589:
12585:
12581:
12578:
12575:
12570:
12567:
12563:
12559:
12554:
12551:
12547:
12542:
12538:
12533:
12506:
12503:
12481:
12477:
12467:
12454:
12448:
12445:
12433:
12430:
12424:
12418:
12406:
12402:
12391:
12387:
12382:
12377:
12372:
12335:
12329:
12326:
12310:
12306:
12295:
12287:
12243:
12240:
12237:
12234:
12224:
12210:
12188:
12184:
12181:
12176:
12172:
12168:
12165:
12162:
12157:
12153:
12149:
12144:
12140:
12135:
12131:
12118:
12103:
12099:
12087:
12085:
12084:Stefan Banach
12081:
12080:
12079:Banach spaces
12074:
12067:Banach spaces
12064:
12061:
12015:
11993:
11989:
11966:
11963:
11957:
11949:
11944:
11929:
11924:
11877:
11869:
11853:
11850:
11837:
11832:
11809:
11775:converges to
11760:
11730:
11727:
11724:
11713:
11709:
11690:
11687:
11684:
11673:
11657:
11651:
11648:
11645:
11634:
11630:
11610:
11587:
11584:
11581:
11578:
11575:
11572:
11564:
11550:
11527:
11512:
11508:
11504:
11502:
11498:
11494:
11490:
11474:
11471:
11464:
11446:
11442:
11421:
11418:
11398:
11395:
11390:
11386:
11382:
11377:
11373:
11364:
11346:
11342:
11338:
11335:
11332:
11327:
11323:
11311:
11300:
11292:
11288:
11277:
11274:
11271:
11267:
11259:
11255:
11251:
11246:
11232:
11212:
11204:
11179:
11154:
11134:
11114:
11062:
11058:
11054:
11038:
11028:
11018:
11005:
11000:
10996:
10990:
10986:
10982:
10977:
10973:
10967:
10963:
10959:
10954:
10950:
10944:
10940:
10936:
10931:
10927:
10921:
10917:
10913:
10910:
10881:
10877:
10861:
10855:
10852:
10849:
10846:
10843:
10840:
10837:
10831:
10780:
10764:
10759:
10755:
10749:
10745:
10741:
10736:
10732:
10726:
10722:
10718:
10713:
10709:
10703:
10699:
10695:
10690:
10686:
10680:
10676:
10672:
10629:
10614:
10610:
10594:
10591:
10580:
10552:
10534:
10516:
10512:
10500:
10485:
10481:
10478:
10475:
10467:
10454:
10438:
10391:
10386:
10359:
10354:
10350:
10344:
10340:
10336:
10333:
10330:
10325:
10321:
10315:
10311:
10307:
10299:
10291:
10280:
10264:
10246:
10242:
10232:
10230:
10226:
10209:
10196:
10183:
10146:
10135:
10086:
10085:inner product
10082:
10076:
10072:
10062:
10046:
10042:
10021:
9999:
9995:
9974:
9969:
9965:
9961:
9956:
9952:
9948:
9945:
9937:
9933:
9929:
9911:
9884:
9863:
9860:
9852:
9851:partial order
9847:
9844:
9838:
9834:
9830:
9819:
9817:
9801:
9790:
9779:
9776:
9765:
9754:
9735:
9732:
9712:
9704:
9688:
9685:
9665:
9662:
9656:
9653:
9650:
9647:
9644:
9637:bilinear map
9636:
9620:
9613:vector space
9612:
9591:
9558:
9543:
9527:
9524:
9521:
9501:
9498:
9495:
9475:
9449:
9444:
9434:
9426:
9421:
9411:
9400:
9398:
9385:
9375:
9370:
9357:
9338:
9333:
9323:
9315:
9310:
9297:
9295:
9282:
9274:
9264:
9259:
9237:
9220:
9209:
9206:
9200:
9189:
9178:
9167:
9164:
9155:
9153:
9137:
9126:
9123:
9099:
9094:
9084:
9079:
9069:
9066:
9063:
9058:
9048:
9043:
9033:
9028:
9018:
9013:
8999:
8983:
8980:
8972:
8964:
8960:
8956:
8943:
8910:
8899:
8849:
8802:
8794:
8778:
8775:
8772:
8765:
8749:
8743:
8740:
8737:
8734:
8731:
8723:
8707:
8687:
8667:
8664:
8661:
8658:
8638:
8635:
8630:
8626:
8622:
8615:
8609:
8599:
8585:
8563:
8559:
8553:
8550:
8547:
8543:
8534:
8531:(also called
8516:
8512:
8506:
8503:
8500:
8496:
8488:
8470:
8466:
8443:
8416:
8409:
8393:
8371:
8368:
8365:
8360:
8355:
8345:
8321:
8317:
8292:
8288:
8282:
8279:
8276:
8272:
8262:
8257:
8255:
8251:
8245:
8241:
8231:
8229:
8210:
8204:
8201:
8197:
8190:
8184:
8181:
8177:
8173:
8165:
8162:(also called
8161:
8157:
8153:
8149:
8133:
8125:
8120:
8117:
8111:
8094:
8091:
8085:
8079:
8071:
8055:
8029:
8025:
8022:
8019:
8006:
8003:
8000:
7971:
7967:
7958:
7954:
7941:
7938:
7935:
7910:
7902:
7887:
7866:
7858:
7842:
7834:
7830:
7826:
7821:
7816:
7812:
7803:
7799:
7793:
7788:
7785:
7782:
7778:
7774:
7768:
7762:
7756:
7736:
7733:
7711:
7707:
7686:
7683:
7680:
7672:
7668:
7664:
7659:
7654:
7650:
7641:
7637:
7633:
7630:
7627:
7619:
7615:
7611:
7606:
7601:
7597:
7588:
7584:
7580:
7574:
7571:
7566:
7563:
7555:
7551:
7547:
7544:
7539:
7535:
7514:
7489:
7481:
7439:
7414:
7397:
7383:
7363:
7340:
7337:
7329:
7315:
7309:
7303:
7297:
7294:
7287:
7271:
7207:
7201:
7198:
7195:
7172:
7166:
7163:
7156:
7151:
7137:
7117:
7095:
7066:
7051:
7036:
7033:
7028:
7018:
7015:
7012:
7007:
6980:
6977:
6966:
6963:
6957:
6951:
6948:
6937:
6934:
6914:
6911:
6907:
6901:
6891:
6886:
6875:
6854:
6851:
6846:
6819:
6816:
6811:
6784:
6742:
6736:
6733:
6725:
6717:
6706:
6703:
6700:
6675:
6668:
6653:
6633:
6629:
6625:
6605:
6602:
6599:
6591:
6586:
6584:
6583:
6566:
6563:
6560:
6552:
6548:
6544:
6540:
6536:
6531:
6517:
6509:
6493:
6473:
6465:
6449:
6429:
6409:
6401:
6385:
6377:
6361:
6319:
6299:
6292:
6284:
6279:
6278:
6272:
6267:
6262:
6258:
6248:
6240:
6237:
6232:
6227:
6222:
6218:
6213:
6207:
6203:
6198:
6194:
6188:
6182:
6177:
6174:, called the
6172:
6155:
6152:
6146:
6143:
6140:
6137:
6134:
6119:
6113:
6107:
6100:
6096:
6092:
6086:
6082:
6077:
6072:
6066:
6062:
6057:
6052:
6047:
6040:
6036:
6032:
6029:
6023:
6016:
6012:
6006:
6000:
5994:
5990:
5986:
5981:
5980:Endomorphisms
5976:
5966:
5964:
5960:
5956:
5951:
5945:
5941:
5940:square matrix
5935:
5930:
5918:
5909:
5900:
5895:
5890:
5886:
5883:
5879:
5875:
5870:
5865:
5859:
5840:
5832:
5816:
5780:
5772:
5768:
5752:
5732:
5728:
5722:
5718:
5712:
5709:
5705:
5699:
5694:
5691:
5688:
5684:
5680:
5677:
5674:
5669:
5665:
5659:
5656:
5652:
5646:
5641:
5638:
5635:
5631:
5627:
5622:
5618:
5612:
5609:
5605:
5599:
5594:
5591:
5588:
5584:
5579:
5567:
5563:
5559:
5556:
5553:
5548:
5544:
5540:
5535:
5531:
5524:
5510:
5504:
5487:
5478:
5472:
5467:
5459:
5454:
5450:
5440:
5437:
5431:
5425:
5419:
5413:
5409:
5405:
5399:
5393:
5387:
5383:-dimensional
5382:
5378:
5374:
5369:
5363:
5357:
5351:
5346:
5341:
5337:
5330:
5324:
5320:
5316:
5310:
5304:
5302:
5297:
5293:
5288:
5283:
5278:
5277:
5271:
5265:
5258:
5254:
5246:
5242:
5234:
5230:
5225:
5217:
5213:
5207:
5204:
5198:
5194:up (down, if
5192:
5186:
5180:
5173:
5169:
5162:
5156:
5151:
5147:
5142:
5137:
5128:
5122:
5116:
5110:
5106:
5103:
5097:
5091:
5085:
5080:
5075:
5069:
5064:
5060:
5055:
5050:
5049:identity maps
5045:
5041:
5037:
5033:
5027:
5023:
5019:
5015:
5011:
5006:
5002:
4998:
4994:
4989:
4985:
4981:
4976:
4975:
4969:
4956:
4953:
4933:
4913:
4910:
4831:
4828:
4825:
4822:
4820:
4807:
4804:
4798:
4791:
4777:
4774:
4760:
4757:
4755:
4742:
4731:
4719:
4715:
4711:
4705:
4695:
4693:
4675:
4671:
4650:
4630:
4610:
4605:
4602:
4598:
4594:
4591:
4588:
4583:
4580:
4576:
4572:
4569:
4563:
4557:
4535:
4532:
4526:
4520:
4517:
4511:
4499:
4495:
4492:
4486:
4471:
4462:
4460:
4441:
4438:
4435:
4429:
4416:
4395:
4376:
4370:
4367:
4364:
4361:
4358:
4311:
4305:
4300:
4295:
4288:
4283:
4278:
4272:
4267:
4264:
4242:
4234:
4226:
4218:
4216:
4200:
4196:
4192:
4189:
4186:
4183:
4180:
4160:
4157:
4153:
4149:
4146:
4143:
4124:
4121:
4097:
4094:
4088:
4083:
4077:
4074:
4068:
4065:
4061:
4054:
4048:
4041:
4038:
4032:
4024:
4021:
4015:
4012:
4008:
4001:
3987:
3981:
3977:
3973:
3963:
3961:
3956:
3952:
3951:integrability
3948:
3943:
3938:
3934:
3930:
3910:
3904:
3898:
3895:
3889:
3883:
3880:
3874:
3865:
3862:
3859:
3833:
3830:
3827:
3815:
3809:
3803:
3775:
3769:
3766:
3763:
3757:
3751:
3748:
3745:
3739:
3730:
3727:
3724:
3688:
3685:
3682:
3679:
3670:
3665:
3655:
3652:
3630:
3625:
3608:
3602:
3596:
3590:
3586:
3585:smaller field
3583:containing a
3581:
3576:
3572:
3567:
3565:
3564:complex plane
3559:
3555:
3548:
3545:
3541:
3536:
3531:
3525:
3519:
3513:
3507:
3500:
3496:
3492:
3488:
3484:
3480:
3476:
3472:
3465:
3461:
3457:
3453:
3449:
3445:
3441:
3437:
3433:
3427:
3422:
3416:
3410:
3406:
3401:
3397:
3391:
3387:
3377:
3375:
3369:
3363:
3359:
3353:
3346:
3341:
3338:and called a
3336:
3330:
3323:
3319:
3297:
3293:
3289:
3286:
3283:
3278:
3274:
3270:
3265:
3261:
3248:
3243:
3240:
3233:
3229:is the field
3227:
3216:
3214:
3209:
3192:
3186:
3183:
3180:
3177:
3174:
3168:
3166:
3158:
3155:
3152:
3146:
3139:
3131:
3127:
3123:
3118:
3114:
3110:
3105:
3101:
3097:
3092:
3088:
3081:
3079:
3069:
3065:
3061:
3056:
3052:
3045:
3037:
3033:
3029:
3024:
3020:
3002:
2998:
2992:
2966:
2963:
2957:
2953:
2946:
2942:
2936:
2929:
2923:
2917:
2914:
2907:
2901:
2898:
2895:
2891:is negative,
2889:
2883:
2880:
2874:
2868:
2863:
2858:
2852:
2846:
2842:
2837:
2833:
2828:
2824:
2823:parallelogram
2819:
2813:
2808:
2804:
2800:
2796:
2784:
2777:
2768:
2756:
2750:
2744:
2740:
2732:
2717:
2707:
2705:
2701:
2699:
2693:
2689:
2685:
2681:
2677:
2673:
2668:
2666:
2661:
2657:
2653:
2650:, as well as
2649:
2645:
2641:
2637:
2633:
2629:
2624:
2622:
2618:
2615:goes back to
2614:
2610:
2606:
2602:
2598:
2594:
2590:
2585:
2583:
2579:
2575:
2571:
2567:
2563:
2559:
2558:Möbius (1827)
2555:
2551:
2547:
2543:
2539:
2535:
2531:
2521:
2519:
2515:
2510:
2504:
2503:componentwise
2478:
2474:
2464:
2459:
2455:
2446:
2441:
2436:
2431:
2413:
2409:
2405:
2402:
2399:
2394:
2390:
2363:
2359:
2355:
2352:
2349:
2344:
2340:
2319:
2314:
2302:
2298:
2294:
2291:
2288:
2283:
2271:
2267:
2263:
2238:
2235:
2219:over a field
2215:of dimension
2193:
2183:
2180:
2177:
2172:
2162:
2157:
2138:
2136:
2132:
2128:
2124:
2119:
2111:
2107:
2103:
2099:
2095:
2091:
2089:
2085:
2082:
2075:
2071:
2070:
2058:
2054:
2013:
2009:
1997:
1995:
1992:
1990:
1988:
1960:
1945:
1941:
1937:
1935:
1932:
1913:
1897:
1895:
1892:
1889:
1871:
1867:
1863:
1860:
1857:
1852:
1848:
1827:
1824:
1821:
1816:
1806:
1803:
1800:
1795:
1768:
1765:
1760:
1756:
1752:
1749:
1746:
1741:
1737:
1716:
1711:
1699:
1695:
1691:
1688:
1685:
1680:
1668:
1664:
1660:
1655:
1643:
1639:
1610:
1608:
1605:
1604:
1596:
1589:
1585:
1580:
1572:
1569:
1562:
1559:
1555:
1549:
1544:
1539:
1533:
1527:
1518:
1516:
1512:
1493:
1485:
1460:
1457:
1454:
1429:
1421:
1414:
1400:
1392:
1389:
1378:
1375:
1365:
1351:
1343:
1335:
1328:
1314:
1306:
1298:
1291:
1290:
1289:
1275:
1272:
1269:
1244:
1241:
1238:
1229:
1216:
1205:
1199:
1191:
1183:
1169:
1167:
1162:
1157:
1153:
1152:abelian group
1148:
1146:
1139:
1125:
1121:
1117:
1113:
1103:
1100:
1096:
1093:
1089:
1085:
1081:
1076:
1073:
1072:
1068:
1065:
1061:
1058:
1054:
1050:
1046:
1042:
1039:
1036:
1035:
1027:
1018:
1014:
1009:
1006:
1005:
1001:
997:
993:
990:
986:
982:
979:
978:
973:
969:
965:
959:
954:
953:
949:, called the
947:
943:
936:
932:
927:
924:
921:
920:
915:
911:
905:
901:
897:
892:
891:
887:, called the
885:
881:
876:
873:
870:
869:
865:
861:
857:
853:
849:
846:
845:Commutativity
843:
842:
838:
834:
830:
826:
822:
818:
814:
811:
810:Associativity
808:
807:
803:
800:
799:
796:
777:
771:
765:
760:
751:
748:
734:
721:
720:
715:
714:
710:
705:
701:
687:
681:
676:
672:
668:
667:
666:
664:
656:
648:
644:
640:
632:
625:
620:
605:
600:
598:
593:
591:
586:
585:
583:
582:
574:
571:
569:
566:
564:
561:
559:
556:
554:
551:
549:
546:
544:
541:
540:
536:
533:
532:
528:
523:
522:
515:
514:
510:
509:
505:
502:
500:
497:
495:
492:
491:
486:
481:
480:
473:
472:
468:
466:
463:
462:
458:
455:
453:
450:
448:
445:
443:
440:
438:
435:
433:
430:
429:
424:
419:
418:
413:
412:
405:
402:
400:
399:Division ring
397:
395:
392:
390:
387:
385:
382:
380:
377:
375:
372:
370:
367:
365:
362:
360:
357:
356:
351:
346:
345:
340:
339:
332:
329:
327:
324:
322:
321:Abelian group
319:
318:
314:
311:
309:
306:
304:
300:
297:
295:
292:
291:
287:
282:
281:
278:
275:
274:
271:
269:
268:Banach spaces
265:
261:
260:normed spaces
257:
253:
249:
245:
241:
237:
233:
228:
226:
222:
218:
214:
210:
206:
202:
198:
194:
190:
186:
181:
179:
175:
171:
167:
164:, but also a
163:
159:
155:
151:
147:
142:
140:
136:
132:
128:
124:
120:
119:
113:
109:
108:
103:
102:
97:
93:
89:
85:
81:
72:
68:
63:
58:
52:
46:
40:
33:
19:
20060:Group theory
20042:
19942:Vector space
19941:
19674:Vector space
19673:
19587:
19541:
19528:, retrieved
19521:the original
19490:
19486:
19457:
19429:
19424:Stewart, Ian
19415:
19380:
19376:
19342:, New York:
19339:
19320:
19285:
19262:, New York:
19259:
19235:
19214:, New York:
19211:
19186:
19164:
19142:
19116:
19110:
19088:
19063:
19038:
19008:
18981:
18975:
18953:
18928:
18903:
18870:
18851:
18814:
18791:
18736:
18715:
18709:
18701:the original
18695:
18681:
18658:
18635:
18626:
18612:
18582:
18576:
18563:
18550:
18546:
18533:
18505:
18499:
18468:
18434:
18413:
18386:
18350:
18328:
18303:
18270:
18249:
18228:
18210:
18192:
18174:
18152:
18130:
18112:
18094:
18082:
18065:
18039:
18017:
17992:
17958:
17946:
17931:, Springer,
17927:
17908:
17886:
17873:
17849:
17830:
17797:
17768:
17749:
17739:
17720:
17693:
17684:
17655:
17635:
17611:
17599:
17577:Coxeter 1987
17572:
17560:
17548:
17541:Grillet 2007
17536:
17524:
17512:
17500:
17488:
17476:
17471:, ch. XVI.8.
17464:
17459:, ch. XVI.7.
17452:
17440:
17428:
17416:
17404:
17399:, Chapter 1.
17392:
17380:
17373:Choquet 1966
17368:
17356:
17344:
17332:
17320:
17308:
17296:
17284:
17272:
17260:
17253:Choquet 1966
17248:
17236:
17224:
17212:
17196:
17184:
17172:
17160:
17148:
17136:
17129:Yoneda lemma
17125:Roman (2005)
17120:
17115:, ch. XVI.1.
17108:
17096:
17084:
17072:
17060:
17055:, ch. IV.3..
17048:
17036:
17024:
17012:
17000:
16988:
16976:
16964:
16952:
16940:
16928:
16916:
16904:
16892:
16880:
16868:
16856:
16844:
16832:
16827:, ch. VI.3..
16820:
16808:
16796:
16784:
16768:
16752:
16736:
16720:
16708:
16696:
16684:
16672:
16660:
16648:
16636:
16624:
16612:
16605:Bolzano 1804
16600:
16588:
16572:
16556:
16540:
16528:
16508:
16496:
16465:
16453:
16441:
16429:
16417:
16397:
16385:
16373:
16361:
16349:
16320:
16315:
16306:
16302:
16297:
16288:
16275:
16202:
16135:
16033:
16016:
15977:Roman (2005)
15971:
15960:affine space
15954:
15941:
15933:
15920:
15879:
15861:
15845:
15844:is known as
15841:
15839:
15834:
15826:
15822:
15817:
15813:
15739:
15735:
15729:
15725:
15720:
15716:
15707:
15703:
15697:
15693:
15688:
15684:
15680:
15594:
15592:
15586:
15582:
15579:affine plane
15565:Affine space
15541:vector space
15540:
15525:free modules
15519:
15515:
15506:
15498:
15494:
15486:
15485:
15455:
15450:
15443:
15436:
15432:
15419:
15408:
15405:vector field
15399:
15395:
15390:
15380:
15370:
15366:
15354:
15351:Möbius strip
15345:
15341:
15336:
15331:
15327:
15323:
15318:
15314:
15310:
15308:neighborhood
15303:
15299:
15298:: for every
15295:
15291:
15283:
15274:
15270:
15264:
15260:
15256:
15251:
15246:is called a
15241:
15236:
15228:
15224:
15188:
15184:
15180:
15173:
15171:
15164:
15160:
14794:
14658:
14404:Lie algebras
14402:
14400:
14349:
14344:
14340:
14336:
14330:
14131:
14087:
14081:
13958:
13920:denotes the
13742:
13740:
13710:
13267:
13151:
12994:
12468:
12088:
12077:
12076:
12073:Banach space
11945:
11711:
11626:
11361:denotes the
11258:infinite sum
11249:
11247:
11030:
10233:
10228:
10224:
10078:
9932:Riesz spaces
9848:
9828:
9825:
9634:
9610:
9540:that maps a
8970:
8968:
8613:
8611:
8535:and denoted
8486:
8260:
8258:
8253:
8249:
8247:
8121:
8115:
8109:
7398:
7152:
6589:
6587:
6580:
6579:is called a
6550:
6546:
6542:
6538:
6534:
6532:
6399:
6375:
6288:
6276:
6246:
6235:
6230:
6225:
6211:
6205:
6201:
6192:
6186:
6180:
6170:
6117:
6111:
6105:
6098:
6094:
6091:identity map
6084:
6080:
6070:
6064:
6060:
6055:
6050:
6045:
6038:
6034:
6030:
6027:
6021:
6014:
6010:
6004:
5998:
5992:
5988:
5984:
5978:
5958:
5954:
5949:
5943:
5933:
5926:
5916:
5907:
5898:
5881:
5877:
5873:
5868:
5863:
5857:
5854:
5508:
5502:
5476:
5470:
5465:
5464:
5435:
5429:
5423:
5417:
5411:
5407:
5403:
5397:
5391:
5385:
5380:
5372:
5367:
5361:
5355:
5349:
5339:
5335:
5328:
5322:
5318:
5314:
5308:
5305:
5300:
5295:
5291:
5281:
5274:
5269:
5263:
5256:
5252:
5244:
5240:
5232:
5228:
5223:
5215:
5211:
5209:Linear maps
5208:
5202:
5196:
5190:
5184:
5178:
5171:
5167:
5160:
5154:
5140:
5133:
5126:
5120:
5114:
5101:
5095:
5089:
5083:
5078:
5073:
5067:
5061:) and onto (
5053:
5043:
5039:
5035:
5031:
5025:
5021:
5017:
5013:
5010:compositions
5004:
5000:
4996:
4987:
4983:
4979:
4972:
4970:
4713:
4709:
4707:
4549:
4458:
4413:denotes the
4256:
3983:
3941:
3813:
3807:
3801:
3794:
3650:
3606:
3600:
3594:
3588:
3579:
3568:
3557:
3553:
3546:
3543:
3539:
3534:
3529:
3523:
3517:
3511:
3505:
3498:
3494:
3490:
3486:
3482:
3478:
3474:
3470:
3463:
3459:
3455:
3451:
3447:
3443:
3439:
3435:
3431:
3420:
3414:
3408:
3405:real numbers
3399:
3395:
3389:
3383:
3373:
3367:
3361:
3357:
3351:
3344:
3342:. The case
3339:
3334:
3328:
3321:
3317:
3315:of elements
3246:
3238:
3231:
3225:
3222:
3210:
3000:
2996:
2991:ordered pair
2972:
2961:
2955:
2951:
2948:. Moreover,
2944:
2940:
2934:
2927:
2921:
2915:
2912:
2905:
2902:
2896:
2893:
2887:
2881:
2878:
2872:
2866:
2861:
2860:, is called
2856:
2850:
2844:
2835:
2831:
2826:
2817:
2811:
2792:
2782:
2775:
2754:
2748:
2742:
2738:
2697:
2669:
2625:
2608:
2604:
2586:
2574:equipollence
2527:
2511:
2462:
2444:
2440:coefficients
2439:
2434:
2429:
2139:
2131:real numbers
2117:
2115:
2105:
2093:
2073:
2069:spanning set
2067:
2056:
2052:
2011:
2007:
1987:intersection
1984:
1943:
1939:
1911:
1888:coefficients
1887:
1840:The scalars
1631:of the form
1611:Given a set
1594:
1587:
1583:
1570:
1567:
1560:
1557:
1553:
1547:
1537:
1531:
1508:
1230:
1170:
1160:
1149:
1141:
1138:vector space
1137:
1123:
1115:
1112:real numbers
1109:
1101:
1098:
1094:
1091:
1087:
1083:
1079:
1066:
1063:
1059:
1056:
1052:
1048:
1044:
1024:denotes the
1016:
1012:
999:
995:
991:
988:
984:
971:
967:
963:
961:, such that
957:
950:
945:
941:
934:
930:
913:
909:
903:
899:
895:
893:, such that
888:
883:
879:
863:
859:
855:
851:
836:
832:
828:
824:
820:
816:
775:
769:
763:
756:
749:
746:
732:
717:
708:
703:
699:
685:
679:
674:
670:
662:
654:
621:
618:
573:Hopf algebra
511:
504:Vector space
503:
469:
409:
338:Group theory
336:
301: /
248:Lie algebras
229:
200:
192:
182:
143:
131:real numbers
126:
122:
115:
106:
100:
92:linear space
91:
88:vector space
87:
77:
70:
66:
61:
56:
50:
32:Vector field
18:Linear space
19922:Multivector
19887:Determinant
19844:Dot product
19689:Linear span
19321:Gravitation
19312:Thorne, Kip
18632:, reprint:
18537:(in German)
18508:: 133–181,
18325:Lang, Serge
18300:Lang, Serge
17794:Lang, Serge
17517:Atiyah 1989
17481:Spivak 1999
17325:Treves 1967
17301:Treves 1967
17289:Treves 1967
17267:, p. 34–36.
17265:Treves 1967
17201:Treves 1967
17031:, ch. 10.4.
16983:, ch. IX.4.
16897:Halmos 1974
16887:, ch. VI.6.
16725:Dorier 1995
16713:Banach 1922
16641:Dorier 1995
16617:Möbius 1827
16577:Halmos 1948
16281:Hamel basis
15964:zero vector
15535:which is a
15441:quaternions
15248:line bundle
15124:yields the
14352:polynomials
14189:Lie algebra
14163:eigenstates
14155:eigenvalues
13713:derivatives
13156:on a given
12715:zero vector
11948:functionals
10607:are called
10263:dot product
9703:composition
8429:an element
7857:derivatives
6539:vector line
6354:-vector of
6289:A nonempty
6051:eigenvector
6025:satisfying
5929:determinant
5871:linear map
5453:Determinant
4993:inverse map
4974:isomorphism
4716:. They are
3984:Systems of
3935:, or other
3799:to a field
3384:The set of
2938:is the sum
2841:real number
2797:in a fixed
2752:(blue) and
2636:linear maps
2601:quaternions
2580:is then an
2534:coordinates
2430:coordinates
2123:Hamel bases
2102:cardinality
2062:, and that
2008:linear span
1994:Linear span
1977:belongs to
890:zero vector
661:are called
558:Lie algebra
543:Associative
447:Total order
437:Semilattice
411:Ring theory
80:mathematics
20049:Categories
19956:Direct sum
19791:Invertible
19694:Linear map
19530:2017-10-25
18763:vct axioms
18423:0070542368
17586:References
17565:Meyer 2000
17553:Meyer 2000
17529:Artin 1991
17435:, ch. 1.6.
17313:Evans 1998
17191:, ch. 1.2.
17189:Naber 2003
17177:Roman 2005
17141:Rudin 1991
17101:Roman 2005
17079:, ch. 7.4.
17065:Roman 2005
17041:Roman 2005
17017:Roman 2005
17005:Roman 2005
16993:Roman 2005
16969:Roman 2005
16935:, ch. V.1.
16921:Roman 2005
16863:, ch. 7.3.
16837:Roman 2005
16803:, ch. V.1.
16791:, ch. I.1.
16775:, p.
16759:, p.
16743:, p.
16729:Moore 1995
16701:Moore 1995
16677:Peano 1888
16579:, p.
16563:, p.
16547:, p.
16545:Joshi 1989
16533:Blass 1984
16519:, p.
16501:Roman 2005
16487:, p.
16470:Brown 1991
16458:Brown 1991
16446:Brown 1991
16410:Brown 1991
16404:, p.
16390:Brown 1991
16378:Roman 2005
16366:Brown 1991
16011:from this.
15376:orientable
15157:looks like
14921:where the
14754:commutator
13591:such that
13225:, denoted
12262:given by
11890:-norm and
10609:orthogonal
8791:is called
8651:or simply
8487:direct sum
8254:direct sum
7474:such that
6582:hyperplane
6231:eigenspace
6217:eigenbasis
6199:, such as
6061:eigenvalue
5961:matrix is
5279:, denoted
5079:isomorphic
5063:surjective
4710:linear map
4704:Linear map
3947:continuity
3577:: a field
3535:isomorphic
2807:velocities
2786:are shown.
2696:spaces of
1579:orthogonal
928:For every
804:Statement
673:or simply
232:structures
189:isomorphic
152:, such as
19986:Numerical
19749:Transpose
19594:EMS Press
19517:0098-3063
19495:CiteSeerX
19397:0010-485X
19377:Computing
18522:0016-2736
18457:840278135
18405:144216834
18379:702357363
18353:, Dover,
18072:EMS Press
17531:, ch. 12.
17469:Lang 2002
17457:Lang 2002
17421:Lang 2002
17409:Lang 1993
17361:Lang 1993
17349:Lang 1993
17327:, ch. 12.
17291:, ch. 11.
17277:Lang 1983
17231:, §4.11-5
17113:Lang 2002
17053:Lang 1987
16957:Lang 1987
16945:Lang 1987
16933:Lang 1987
16909:Lang 1987
16885:Lang 1987
16873:Lang 1987
16849:Lang 1987
16825:Lang 1987
16813:Lang 1993
16801:Lang 2002
16789:Lang 1987
16773:Jain 2001
16679:, ch. IX.
16561:Heil 2011
16513:Lang 1987
16354:Lang 2002
16342:Citations
16255:Ω
16217:≠
16100:‖
16093:‖
16081:‖
16074:‖
16071:≤
16062:‖
16049:‖
15897:→
15831:nullspace
15711:(it is a
15651:↦
15622:→
15616:×
15593:Roughly,
15448:octonions
15208:→
15199:π
15100:⊗
15085:−
15067:⊗
15016:⊗
14972:⊗
14894:⊗
14891:⋯
14888:⊗
14873:⊗
14809:
14731:−
14508:−
14388:quotients
14282:−
14276:⋅
14212:⋅
14199:hyperbola
13906:¯
13858:¯
13825:Ω
13821:∫
13814:⟩
13796:⟨
13770:Ω
13715:leads to
13681:μ
13643:−
13620:Ω
13616:∫
13610:∞
13607:→
13576:Ω
13496:μ
13458:−
13428:Ω
13424:∫
13418:∞
13415:→
13375:∞
13363:‖
13349:‖
13329:…
13310:…
13248:Ω
13206:∞
13194:‖
13187:‖
13167:Ω
13109:μ
13070:Ω
13066:∫
13047:‖
13040:‖
13012:→
13009:Ω
12970:−
12962:⋅
12941:−
12909:∑
12896:‖
12880:‖
12854:→
12846:−
12821:−
12802:∞
12798:‖
12782:‖
12731:∞
12673:−
12613:…
12590:−
12579:…
12568:−
12552:−
12478:ℓ
12452:∞
12388:∑
12369:‖
12360:‖
12333:∞
12283:∞
12279:‖
12270:‖
12247:∞
12244:≤
12238:≤
12185:…
12166:…
12100:ℓ
12019:→
11994:∗
11961:→
11910:-norm on
11898:∞
11838:−
11816:∞
11813:→
11608:∞
11399:…
11336:⋯
11318:∞
11315:→
11283:∞
11268:∑
10914:−
10908:⟩
10890:⟨
10807:⟩
10789:⟨
10742:−
10670:⟩
10652:⟨
10589:⟩
10573:⟨
10535:⋅
10517:⋅
10490:∠
10482:
10468:⋅
10334:⋯
10300:⋅
10289:⟩
10273:⟨
10205:⟩
10189:⟨
10144:⟩
10128:⟨
10047:−
9970:−
9962:−
9861:≤
9833:converges
9766:⊗
9660:→
9654:×
9592:⊗
9525:⊗
9499:×
9488:from the
9435:⊗
9412:⊗
9358:⊗
9339:⊗
9316:⊗
9283:⊗
9210:⋅
9201:⊗
9179:⊗
9168:⋅
9138:⊗
9127:⋅
9085:⊗
9067:⋯
9049:⊗
9019:⊗
8971:universal
8897:↦
8776:×
8762:from the
8747:→
8741:×
8662:⊗
8627:⊗
8551:∈
8544:∐
8533:coproduct
8504:∈
8497:⨁
8408:index set
8369:∈
8280:∈
8273:∏
8205:
8198:≡
8185:
8034:′
8026:⋅
8015:′
8004:⋅
7976:′
7963:′
7950:′
7895:′
7892:′
7779:∑
7760:↦
7631:⋯
7412:↦
7338:∈
7298:
7205:→
7167:
6967:⋅
6938:⋅
6734:∈
6603:⊆
6564:−
6541:), and a
6144:⋅
6141:λ
6138:−
5830:↦
5767:summation
5753:∑
5685:∑
5678:…
5632:∑
5585:∑
5576:↦
5557:…
5087:are, via
5059:injective
4829:⋅
4808:⋅
4718:functions
4603:−
4581:−
4504:′
4479:′
4476:′
4187:−
3929:real line
3849:given by
3770:
3752:
3697:→
3287:…
2648:dimension
2640:Grassmann
2626:In 1857,
2403:…
2353:…
2292:⋯
2236:∈
2181:…
2106:dimension
2088:dimension
2057:generates
1861:…
1822:∈
1804:…
1766:∈
1750:…
1689:⋯
1529:A vector
1393:−
1376:−
1270:∈
1242:∈
1206:−
1184:−
1164:into the
568:Bialgebra
374:Near-ring
331:Lie group
299:Semigroup
223:have the
217:countably
185:dimension
166:direction
162:magnitude
20032:Category
19971:Subspace
19966:Quotient
19917:Bivector
19831:Bilinear
19773:Matrices
19648:Glossary
19568:36131259
19540:(1994).
19426:(1975),
19414:(1999),
19318:(1973),
19283:(1998),
19210:(1991),
19087:(1974),
19062:(1995),
19007:(1995),
18952:(1987),
18927:(1989),
18902:(1998),
18850:(1969),
18815:K-theory
18813:(1989),
18788:(1976),
18745:citation
18735:(1888),
18693:(1827),
18679:(1853),
18624:(1844),
18610:(1822),
18562:(1969),
18531:(1804),
18491:(1922),
18467:(1967),
18327:(1993),
18302:(1983),
18269:(1989),
18248:(1988),
18083:Topology
18081:(1966),
18016:(2004),
17991:(1987),
17981:Analysis
17957:(1993),
17885:(2005),
17829:(1999),
17796:(2002),
17683:(1948),
17610:(1991),
17483:, ch. 3.
17339:, p.190.
17315:, ch. 5.
17243:, §1.5-5
17179:, ch. 9.
16689:Guo 2021
16128:seminorm
15850:parallel
15549:spectrum
15466:Sections
15446:and the
15424:K-theory
15417:2-sphere
15363:cylinder
11633:complete
11248:In such
11053:topology
8884:the map
8793:bilinear
8406:in some
8152:category
8146:) is an
7110:lies in
6400:subspace
6043:, where
5987: :
5876: :
5765:denotes
5466:Matrices
5443:Matrices
5406: :
5317: :
5038: :
5020: :
4999: :
4982: :
4859:for all
4215:Matrices
3933:interval
2710:Examples
2656:algebras
2617:Laguerre
2597:Hamilton
2544:founded
1957:that is
1447:implies
1288:one has
1020:, where
907:for all
675:addition
404:Lie ring
369:Semiring
236:algebras
209:geometry
174:matrices
158:velocity
19643:Outline
19596:, 2001
19560:1269324
19405:9738629
19362:2044239
19135:2035388
19031:1322960
18998:2320587
18893:0763890
18837:1043170
18601:1347828
18293:0992618
17959:Algebra
17911:, CRC,
17877:, Lyryx
17831:Algebra
17820:1878556
17798:Algebra
17612:Algebra
17591:Algebra
17219:, p. 7.
16325:section
16296:from π(
15531:over a
15489:are to
15487:Modules
15476:Modules
15288:locally
15278:into a
14852:tensors
14083:closure
12036:(or to
10451:by the
9725:equals
8998:tensors
5480:matrix
5287:natural
4690:is the
4550:yields
3937:subsets
3927:is the
3562:in the
3424:is the
3242:-tuples
2885:. When
2688:algebra
2684:Hilbert
2554:Bolzano
2524:History
2493:of the
1581:basis:
1513:over a
663:scalars
655:vectors
535:Algebra
527:Algebra
432:Lattice
423:Lattice
116:vector
107:scalars
101:vectors
94:) is a
84:physics
19927:Tensor
19739:Kernel
19669:Vector
19664:Scalar
19566:
19558:
19548:
19515:
19497:
19468:
19446:
19403:
19395:
19360:
19350:
19328:
19297:
19270:
19248:
19222:
19197:
19175:
19153:
19133:
19099:
19074:
19049:
19029:
19019:
18996:
18964:
18939:
18914:
18891:
18881:
18835:
18825:
18800:
18666:
18646:
18599:
18520:
18455:
18445:
18420:
18403:
18393:
18377:
18367:
18339:
18314:
18291:
18281:
18256:
18235:
18217:
18199:
18181:
18163:
18141:
18119:
18101:
18050:
18028:
18003:
17969:
17935:
17915:
17897:
17861:
17837:
17818:
17808:
17783:
17757:
17728:
17708:
17670:
17644:
17622:
17143:, p.3.
15821:where
15603:action
15589:(red).
15529:module
15482:Module
15439:, the
15231:, the
14923:degree
14539:), and
14165:. The
13882:where
13808:
13802:
13409:
13158:domain
12202:whose
11565:, for
11563:-norms
11491:. The
11304:
11298:
11254:series
11127:, and
9987:where
9829:per se
9404:
9394:
9301:
9291:
9230:
9227:
9193:
9187:
9159:
9149:
8228:groups
7155:kernel
6755:where
6667:modulo
6549:is an
6537:(also
6291:subset
6283:planes
6271:origin
6076:kernel
5914:, and
5745:where
5449:Matrix
5433:, via
5338:= dim
5301:bidual
5158:- and
5146:origin
4623:where
4417:, and
4257:where
3978:, and
3931:or an
3592:is an
3418:where
2821:, the
2803:forces
2795:arrows
2680:Banach
2628:Cayley
2593:Argand
2006:, the
1985:every
1959:closed
1729:where
1601:(red).
1511:module
801:Axiom
783:, and
759:axioms
647:axioms
641:and a
633:
563:Graded
494:Module
485:Module
384:Domain
303:Monoid
154:forces
118:axioms
19796:Minor
19781:Block
19719:Basis
19524:(PDF)
19483:(PDF)
19401:S2CID
19131:JSTOR
18994:JSTOR
18867:(PDF)
18496:(PDF)
16300:) to
15872:Notes
15866:flags
15713:coset
15537:field
15491:rings
15233:fiber
14088:basis
13341:with
12872:but
12223:-norm
11363:limit
9705:with
9542:tuple
7286:image
6543:plane
6103:. If
6059:with
5938:of a
5932:det (
5377:up to
5249:, or
3714:with
3481:) = (
3446:) = (
3438:) + (
2799:plane
2660:Peano
2550:curve
2454:tuple
2332:with
2118:Bases
2094:basis
2084:Basis
2072:or a
2066:is a
2053:spans
1902:of a
1515:field
1140:or a
994:) = (
827:) = (
624:field
529:-like
487:-like
425:-like
394:Field
352:-like
326:Magma
294:Group
288:-like
286:Group
139:field
19951:Dual
19806:Rank
19564:OCLC
19546:ISBN
19513:ISSN
19466:ISBN
19444:ISBN
19393:ISSN
19348:ISBN
19326:ISBN
19295:ISBN
19268:ISBN
19246:ISBN
19220:ISBN
19195:ISBN
19173:ISBN
19151:ISBN
19097:ISBN
19072:ISBN
19047:ISBN
19017:ISBN
18962:ISBN
18937:ISBN
18912:ISBN
18879:ISBN
18823:ISBN
18798:ISBN
18765:via
18751:link
18664:ISBN
18644:ISBN
18518:ISSN
18453:OCLC
18443:ISBN
18418:ISBN
18401:OCLC
18391:ISBN
18375:OCLC
18365:ISBN
18337:ISBN
18312:ISBN
18279:ISBN
18254:ISBN
18233:ISBN
18215:ISBN
18197:ISBN
18179:ISBN
18161:ISBN
18139:ISBN
18117:ISBN
18099:ISBN
18048:ISBN
18026:ISBN
18001:ISBN
17967:ISBN
17933:ISBN
17913:ISBN
17895:ISBN
17859:ISBN
17855:SIAM
17835:ISBN
17806:ISBN
17781:ISBN
17755:ISBN
17726:ISBN
17706:ISBN
17668:ISBN
17642:ISBN
17620:ISBN
16206:For
16041:for
16037:The
15864:and
15856:and
15733:for
15567:and
15533:ring
15456:The
15357:(by
15145:and
14996:and
14795:The
14752:the
14679:-by-
14459:and
14339:(or
14233:The
14187:and
13372:<
13203:<
12717:for
12449:<
11600:and
11499:and
11197:and
11085:and
10426:and
10227:and
10119:and
10081:norm
10073:and
10034:and
9633:and
8837:and
8700:and
8612:The
8259:The
8248:The
8242:and
7990:and
7855:the
7376:and
7153:The
7081:and
6832:and
6535:line
6464:span
6259:and
6087:· Id
5957:-by-
5927:The
5861:and
5474:-by-
5451:and
5353:and
5343:, a
5334:dim
5289:map
5124:and
5071:and
5047:are
5029:and
4926:all
4881:and
4643:and
4173:and
3811:and
3654:.
3527:and
3489:) +
3468:and
3454:) +
3412:and
3403:for
3372:(so
3365:and
2985:and
2977:and
2950:(−1)
2815:and
2779:and
2702:and
2682:and
2646:and
2607:and
2595:and
2576:. A
2540:and
2512:The
2086:and
2012:span
1781:and
1257:and
1055:) =
970:) =
966:+ (−
835:) +
787:and
773:and
683:and
359:Ring
350:Ring
266:and
246:and
215:are
156:and
133:and
125:and
86:, a
82:and
19505:doi
19385:doi
19121:doi
18986:doi
18720:doi
18587:doi
18510:doi
18439:GTM
18355:hdl
17773:doi
17698:doi
17660:doi
16745:355
16565:126
16549:450
16521:212
16406:185
15837:).
15833:of
15719:in
15715:of
15679:If
15577:An
15407:on
15313:of
15302:in
15244:= 1
15227:in
14479:):
14141:in
14090:of
13924:of
13600:lim
13399:lim
13274:not
12810:sup
12292:sup
11806:lim
11712:all
11513:in
11308:lim
11147:in
11107:in
10479:cos
10372:In
9635:any
9611:any
9579:to
9514:to
8795:if
8458:of
8182:ker
8119:).
8113:or
7264:in
7164:ker
6867:is
6402:of
6378:of
6178:of
6129:det
6053:of
5869:any
5506:to
5439:.
5427:to
5267:to
5251:𝓛(
5221:Hom
4971:An
4946:in
4903:in
4712:or
3953:or
3939:of
3767:exp
3749:sin
3731:exp
3725:sin
3686:exp
3680:sin
3473:⋅ (
3370:= 2
3347:= 1
3326:of
3250:)
2954:= −
2908:= 2
2870:by
2864:of
2827:sum
2805:or
2674:by
2591:by
2460:of
2432:of
2378:in
2076:of
2055:or
2040:If
2014:of
1942:or
1545:of
1535:in
1473:or
1028:in
955:of
819:+ (
791:in
779:in
736:in
726:in
709:sum
689:in
631:set
364:Rng
96:set
78:In
69:+ 2
20051::
19592:,
19586:,
19562:.
19556:MR
19554:.
19511:,
19503:,
19491:38
19489:,
19485:,
19464:,
19460:,
19442:,
19434:,
19399:,
19391:,
19371:;
19358:MR
19356:,
19346:,
19314:;
19310:;
19293:,
19266:,
19244:,
19193:,
19171:,
19149:,
19129:,
19117:17
19115:,
19095:,
19070:,
19045:,
19027:MR
19025:,
19015:,
18992:,
18982:86
18980:,
18960:,
18935:,
18910:,
18889:MR
18887:,
18869:,
18846:;
18833:MR
18831:,
18821:,
18784:;
18761::
18747:}}
18743:{{
18716:22
18714:,
18642:,
18597:MR
18595:,
18583:22
18581:,
18575:,
18551:13
18549:,
18516:,
18498:,
18451:.
18437:.
18399:.
18373:,
18363:,
18335:,
18310:,
18306:,
18289:MR
18287:,
18277:,
18159:,
18137:,
18133:,
18070:,
18064:,
18046:,
18024:,
17999:,
17965:,
17893:,
17857:,
17853:,
17816:MR
17814:,
17800:,
17779:,
17704:,
17666:,
17618:,
17614:,
17203:;
16777:11
16727:;
16581:12
16489:14
16477:^
16408:;
16305:×
16024:,
15934:by
15816:+
15738:∈
15728:+
15706:+
15696:∈
15518:/2
15513:)
15472:.
15464:.
15453:.
15435:,
15431::
15398:×
15369:×
15344:×
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4823:=
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2303:n
2299:a
2295:+
2289:+
2284:1
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2272:1
2268:a
2264:=
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2239:V
2232:v
2221:F
2217:n
2213:V
2199:)
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2189:b
2184:,
2178:,
2173:2
2168:b
2163:,
2158:1
2153:b
2148:(
2080:.
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2064:G
2060:W
2050:G
2046:G
2042:W
2038:.
2036:G
2032:G
2028:G
2024:G
2020:V
2016:G
2004:V
2000:G
1979:W
1975:W
1971:W
1967:W
1963:W
1955:V
1951:V
1947:W
1928:V
1924:G
1920:G
1916:G
1908:V
1904:F
1900:G
1872:k
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1548:R
1538:R
1532:v
1494:.
1490:0
1486:=
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1434:0
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1422:s
1401:,
1397:v
1390:=
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1379:1
1373:(
1352:,
1348:0
1344:=
1340:0
1336:s
1315:,
1311:0
1307:=
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1299:0
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1266:v
1245:F
1239:s
1217:.
1214:)
1210:w
1203:(
1200:+
1196:v
1192:=
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1180:v
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1144:F
1135:-
1133:F
1128:F
1102:v
1099:b
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910:v
904:v
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896:v
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880:0
864:u
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856:v
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753:.
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733:v
728:F
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691:V
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635:V
627:F
603:e
596:t
589:v
74:.
71:w
67:v
62:w
57:w
51:v
41:.
34:.
20:)
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