6997:. In the field of real or complex numbers, the distance of the discrete metric from zero is not homogeneous in the non-zero point; indeed, the distance from zero remains one as its non-zero argument approaches zero. However, the discrete distance of a number from zero does satisfy the other properties of a norm, namely the triangle inequality and positive definiteness. When applied component-wise to vectors, the discrete distance from zero behaves like a non-homogeneous "norm", which counts the number of non-zero components in its vector argument; again, this non-homogeneous "norm" is discontinuous.
15138:
14423:
6432:
9528:
7470:
4005:
6152:
5994:
7255:
7938:
5379:
6422:
9664:
topology precisely when the seminorm can distinguish between distinct vectors, which is again equivalent to the seminorm being a norm. The topology thus defined (by either a norm or a seminorm) can be understood either in terms of sequences or open sets. A
11132:
6335:
6677:
3595:
10736:
3839:
9439:
6024:
4784:
5573:
4271:
2708:
4073:
9964:
5176:
12096:, Mathematics and its Applications (East European Series), vol. 29 (Translated from the Polish by Ewa Bednarczuk ed.), Dordrecht; Warsaw: D. Reidel Publishing Co.; PWN—Polish Scientific Publishers, pp. xvi, 524,
5858:
4176:
7116:. Indeed, it is not even an F-norm in the sense described above, since it is discontinuous, jointly and severally, with respect to the scalar argument in scalar–vector multiplication and with respect to its vector argument.
2567:
4363:
10538:
10741:
Equivalent norms define the same notions of continuity and convergence and for many purposes do not need to be distinguished. To be more precise the uniform structure defined by equivalent norms on the vector space is
10205:
9523:
4984:
2483:
8524:
9220:
7797:
5248:
10614:
10381:
6340:
3657:
10969:
10461:
6848:
5832:. However, aside from trivial cases, this topological vector space is not locally convex, and has no continuous non-zero linear forms. Thus the topological dual space contains only the zero functional.
1457:
6578:
5758:
11002:
1809:
6243:
6583:
3466:
7659:
5115:
4672:
2636:
but there are other norms on this vector space as will be shown below. However, all these norms are equivalent in the sense that they all define the same topology on finite-dimensional spaces.
10619:
10738:
If the vector space is a finite-dimensional real or complex one, all norms are equivalent. On the other hand, in the case of infinite-dimensional vector spaces, not all norms are equivalent.
1996:
8419:
1898:
13830:
9776:
7507:
9337:
13613:
1315:
7465:{\displaystyle \|x\|_{p}={\bigg (}\sum _{i\in \mathbb {N} }\left|x_{i}\right|^{p}{\bigg )}^{1/p}{\text{ and }}\ \|f\|_{p,X}={\bigg (}\int _{X}|f(x)|^{p}~\mathrm {d} x{\bigg )}^{1/p}}
13703:
6951:
4677:
3384:
7540:
4205:
3150:
9999:
6878:
4522:
2156:
433:
10032:
9095:
1593:
875:
303:
5474:
5075:
3079:
2918:
12080:
12048:
10269:
7970:
6471:
5465:
4225:
4095:
3834:
3772:
3736:
3006:
2662:
2634:
2406:
2310:
139:
has been used for several related meanings. It may be a synonym of "seminorm". A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "
11561:
8209:
8173:
8144:
7792:
4020:
13740:
8943:
7571:
1401:
9847:
7602:
9805:
9019:
8995:
8971:
5421:
5201:
3682:
3338:
3310:
3282:
3257:
2225:
533:
242:
11392:
11233:
10237:
9041:
8683:
8449:
6502:
5649:
5610:
3704:
3457:
3428:
3406:
2282:
2200:
2178:
13779:
13547:
8790:
1686:
13906:
7248:
5235:
3179:
998:
9700:
8307:
7756:
7726:
7695:
7189:
4612:
4466:
4432:
2878:
2772:
468:
159:
10993:
9252:
6206:
5814:
2488:
11453:
9631:
8741:
8053:
4887:
4565:
1248:
1027:
752:
723:
622:
10126:
6172:
3114:
1642:
1219:
934:
786:
657:
559:
11516:
10854:
9327:
7216:
7145:
5676:
2848:
2810:
2742:
1925:
8627:
6238:
5040:
4813:
4288:
3207:
3035:
812:
683:
11330:
11281:
11169:
10465:
6732:
4843:
2060:
1740:
333:
11304:
10883:
10827:
9727:
9118:
8841:
8709:
8350:
8021:
7066:
6901:
6702:
6019:
4398:
4000:{\displaystyle \|{\boldsymbol {z}}\|:={\sqrt {\left|z_{1}\right|^{2}+\cdots +\left|z_{n}\right|^{2}}}={\sqrt {z_{1}{\bar {z}}_{1}+\cdots +z_{n}{\bar {z}}_{n}}}.}
2333:
2019:
1616:
1551:
1488:
1109:
582:
11489:
11416:
11354:
11189:
10804:
10778:
10100:
10080:
10052:
9651:
9597:
9296:
9272:
8881:
8861:
8818:
8374:
8327:
8274:
8250:
8093:
8073:
7998:
7106:
7086:
7043:
5853:
5778:
5004:
4907:
4867:
4585:
3799:
2939:
2371:
2115:
2095:
1945:
1833:
1706:
1528:
1508:
1355:
1335:
1193:
1173:
1149:
1129:
1086:
1066:
954:
899:
841:
357:
266:
217:
194:
10130:
8590:
8557:
9448:
6983:
takes the value one for distinct points and zero otherwise. When applied coordinate-wise to the elements of a vector space, the discrete distance defines the
4920:
6147:{\displaystyle {\frac {\partial \|\mathbf {x} \|_{p}}{\partial \mathbf {x} }}={\frac {\mathbf {x} \circ |\mathbf {x} |^{p-2}}{\|\mathbf {x} \|_{p}^{p-1}}}.}
9852:
12657:
9127:
5120:
14459:
10542:
10274:
4103:
10888:
14312:
13937:
12991:
10388:
13636:
124:
satisfies the first two properties of a norm, but may be zero for vectors other than the origin. A vector space with a specified norm is called a
14148:
13113:
5681:
14969:
12645:
11665: – nonnegative-real-valued function on a real or complex vector space that satisfies the triangle inequality and is absolutely homogenous
2411:
13975:
13932:
5989:{\displaystyle {\frac {\partial }{\partial x_{k}}}\|\mathbf {x} \|_{p}={\frac {x_{k}\left|x_{k}\right|^{p-2}}{\|\mathbf {x} \|_{p}^{p-1}}}.}
8454:
3600:
3227:
1032:
Some authors include non-negativity as part of the definition of "norm", although this is not necessary. Although this article defined "
14586:
14561:
14138:
13046:
14543:
14265:
14120:
13018:
6752:
15011:
14513:
14452:
14096:
12652:
12484:
11236:
11138:
1406:
161:" in the homogeneity axiom. It can also refer to a norm that can take infinite values, or to certain functions parametrised by a
14756:
14580:
13365:
6507:
1252:
1753:
13307:
12445:
12415:
12385:
12324:
12293:
11978:
11824:
11707:
7933:{\displaystyle \|x\|:=2\left|x_{1}\right|+{\sqrt {3\left|x_{2}\right|^{2}+\max(\left|x_{3}\right|,2\left|x_{4}\right|)^{2}}}}
5374:{\displaystyle \langle \left(x_{n}\right)_{n},\left(y_{n}\right)_{n}\rangle _{\ell ^{2}}~=~\sum _{n}{\overline {x_{n}}}y_{n}}
6417:{\displaystyle {\frac {\partial }{\partial \mathbf {x} }}\|\mathbf {x} \|_{2}={\frac {\mathbf {x} }{\|\mathbf {x} \|_{2}}}.}
13123:
13206:
4479:
2074:
over themselves and a two-dimensional vector space over the reals; the absolute value is a norm for these two structures.
15021:
14518:
14488:
12673:
5083:
4617:
15141:
14792:
14445:
13988:
12351:
7613:
2021:
There are also a large number of norms that exhibit additional properties that make them useful for specific problems.
1950:
14929:
14077:
13968:
12847:
12630:
12258:
12192:
12149:
12109:
8379:
1838:
13795:
9603:
with congruent axes (see the accompanying illustration). Due to the definition of the norm, the unit circle must be
7068:
or the
Hamming distance of the vector from zero. When this "norm" is localized to a bounded set, it is the limit of
1553:
Any two norms on a finite-dimensional space are equivalent but this does not extend to infinite-dimensional spaces.
14834:
14347:
13442:
12765:
12608:
12250:
13992:
13295:
13231:
12782:
11602:
11127:{\displaystyle \left\{x\in X:p_{A}(x)<1\right\}~\subseteq ~A~\subseteq ~\left\{x\in X:p_{A}(x)\leq 1\right\}.}
9732:
7477:
13076:
9607:
and centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle, and
7113:
14864:
13569:
13487:
13290:
12969:
12748:
6330:{\displaystyle {\frac {\partial }{\partial x_{k}}}\|\mathbf {x} \|_{2}={\frac {x_{k}}{\|\mathbf {x} \|_{2}}},}
4406:, which has dimension equal to the dimension of the vector space minus 1. The Taxicab norm is also called the
14996:
14598:
14575:
14143:
13666:
13328:
13106:
13061:
13051:
12377:
11653:
9442:
8026:
7120:, some engineers omit Donoho's quotation marks and inappropriately call the number-of-non-zeros function the
6906:
6672:{\displaystyle \|\mathbf {x} \|_{\infty }:=\max \left(\left|x_{1}\right|,\ldots ,\left|x_{n}\right|\right).}
3590:{\displaystyle \lVert q\rVert ={\sqrt {\,qq^{*}~}}={\sqrt {\,q^{*}q~}}={\sqrt {\,a^{2}+b^{2}+c^{2}+d^{2}~}}}
3343:
2335:
which exists since such a vector is obtained by multiplying any non-zero vector by the inverse of its norm.
15172:
15162:
15047:
14426:
14199:
14133:
13961:
13163:
13153:
13081:
13008:
12884:
12553:
12281:
7516:
6957:-norm described above is not a norm in the usual sense because it lacks the required homogeneity property.
4181:
3119:
13158:
10731:{\displaystyle \|x\|_{\infty }\leq \|x\|_{2}\leq \|x\|_{1}\leq {\sqrt {n}}\|x\|_{2}\leq n\|x\|_{\infty }.}
9971:
6857:
6741:
In probability and functional analysis, the zero norm induces a complete metric topology for the space of
4914:
2124:
373:
14868:
14163:
13501:
13491:
13091:
12477:
10004:
9571:(the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a
9121:
9066:
8747:
8527:
6175:
1564:
846:
274:
17:
5045:
2888:
877:
that has properties (1.) and (2.) so that in particular, every norm is also a seminorm (and thus also a
15104:
14641:
14556:
14551:
14493:
14408:
14362:
14286:
14168:
13860:
13662:
13324:
13138:
13031:
13026:
12921:
12894:
12859:
12711:
12604:
12053:
12021:
10242:
8095:
applied to the taxicab norm, up to inversion and interchanging of axes, gives a different unit ball: a
7943:
6437:
5468:
5429:
4078:
3807:
3745:
3709:
3040:
2979:
2607:
2379:
2287:
11521:
8185:
8149:
8120:
7768:
4218:, including Euclidean and complex spaces. For complex spaces, the inner product is equivalent to the
1708:
is only a seminorm. For the length of a vector in
Euclidean space (which is an example of a norm, as
14900:
14710:
14403:
14219:
13718:
13618:
13312:
13285:
13268:
13086:
12931:
12600:
9043:
the composition algebra norm is the square of the norm discussed above. In those cases the norm is a
8886:
7549:
1368:
9814:
9332:
8075:
a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. Each
7576:
14673:
14668:
14661:
14656:
14528:
14468:
14255:
14153:
14056:
13835:
13128:
13118:
13036:
12974:
12901:
12855:
12770:
12595:
11145:
consisting of absolutely convex sets. A common method to construct such a basis is to use a family
9781:
9434:{\displaystyle |\langle x,y\rangle |\leq \|x\|_{p}\|y\|_{q}\qquad {\frac {1}{p}}+{\frac {1}{q}}=1.}
9052:
8999:
8975:
8951:
8712:
5817:
5384:
5181:
3802:
3662:
3318:
3290:
3262:
3237:
2205:
2071:
478:
222:
39:
11899:
11359:
11198:
10210:
9552:
9024:
8644:
8428:
6485:
5619:
5580:
3687:
3440:
3411:
3389:
2230:
2183:
2161:
2117:
is equivalent (up to scaling) to the absolute value norm, meaning that there is a norm-preserving
15167:
14934:
14915:
14591:
14571:
14352:
14128:
13757:
13520:
13101:
13041:
11614:
10062:
if they induce the same topology, which happens if and only if there exist positive real numbers
9044:
8750:
1647:
13869:
7223:
6426:
5210:
3155:
968:
15123:
15113:
15097:
14797:
14746:
14646:
14631:
14383:
14327:
14291:
13143:
13056:
12837:
12753:
12589:
12583:
12470:
9672:
8282:
7731:
7704:
7673:
7167:
4779:{\displaystyle \|\mathbf {x} \|_{p}:=\left(\sum _{i=1}^{n}\left|x_{i}\right|^{p}\right)^{1/p}.}
4590:
4444:
4410:
2962:
2856:
2750:
2644:
438:
142:
90:
62:
13383:
10974:
9225:
7670:
Generally, these norms do not give the same topologies. For example, an infinite-dimensional
6181:
5783:
15092:
14779:
14761:
14726:
14566:
14090:
13927:
13922:
13397:
13345:
13302:
13226:
13179:
12916:
12578:
12545:
12518:
11837:
11816:
11747:
11425:
9610:
8717:
8113:
There are examples of norms that are not defined by "entrywise" formulas. For instance, the
7978:
5424:
5204:
4872:
4544:
1224:
1003:
881:). However, there exist seminorms that are not norms. Properties (1.) and (2.) imply that if
728:
690:
598:
586:
14086:
13216:
11700:
10105:
6157:
3084:
1621:
1198:
904:
756:
627:
538:
15108:
15052:
15031:
14366:
13865:
13071:
13066:
12777:
12661:
12567:
12403:
12119:
11632:
11608:
11494:
10858:
10832:
10743:
9807:
Equivalently, the topology consists of all sets that can be represented as a union of open
9703:
9305:
8686:
8114:
7194:
7123:
7117:
5654:
5568:{\displaystyle \langle f,g\rangle _{L^{2}}=\int _{X}{\overline {f(x)}}g(x)\,\mathrm {d} x.}
2826:
2788:
2720:
1903:
1464:
472:
363:
269:
70:
13953:
8603:
6214:
5016:
4789:
4266:{\displaystyle \|{\boldsymbol {x}}\|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.}
3188:
3081:(as first suggested by Euler) the Euclidean norm associated with the complex number. For
3011:
2703:{\displaystyle \|{\boldsymbol {x}}\|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.}
791:
662:
8:
14991:
14986:
14944:
14523:
14332:
14270:
13984:
13708:
13509:
13466:
13280:
13003:
12733:
12540:
11309:
11260:
11240:
11148:
8630:
8214:
There are also norms on spaces of matrices (with real or complex entries), the so-called
7152:
6742:
6711:
5821:
5613:
4822:
4219:
4215:
4208:
4068:{\displaystyle \|{\boldsymbol {x}}\|:={\sqrt {{\boldsymbol {x}}^{H}~{\boldsymbol {x}}}},}
2952:
2582:
2035:
1715:
1460:
367:
308:
125:
78:
69:
to the non-negative real numbers that behaves in certain ways like the distance from the
13654:
11286:
10865:
10809:
9709:
9544:
9100:
8823:
8691:
8332:
8003:
7048:
6883:
6684:
6001:
4380:
2315:
2001:
1598:
1533:
1470:
1091:
564:
14976:
14919:
14853:
14838:
14705:
14357:
14224:
13942:
13853:
13476:
13446:
13263:
13221:
12828:
12738:
12530:
12369:
11671:
11626:
11474:
11401:
11339:
11174:
10789:
10763:
10085:
10065:
10037:
9808:
9636:
9582:
9548:
9281:
9257:
8866:
8846:
8803:
8359:
8353:
8312:
8277:
8259:
8235:
8078:
8058:
7983:
7975:
7091:
7071:
7028:
6994:
5838:
5825:
5763:
4989:
4892:
4852:
4570:
4437:
3784:
2924:
2356:
2348:
2100:
2080:
1930:
1818:
1691:
1513:
1493:
1362:
1358:
1340:
1320:
1178:
1158:
1134:
1114:
1071:
1051:
939:
884:
878:
826:
342:
251:
202:
179:
82:
13784:
13754:
13715:
8563:
8560:, the Galois-theoretic norm is not a norm in the sense of this article. However, the
8530:
14688:
14614:
14337:
13623:
13096:
12877:
12820:
12800:
12451:
12441:
12421:
12411:
12391:
12381:
12357:
12347:
12330:
12320:
12299:
12289:
12264:
12254:
12188:
12145:
12123:
12105:
11974:
11820:
11793:
11703:
11461:
10781:
9959:{\displaystyle \|x-y\|=\|x-z\|+\|z-y\|{\text{ for all }}x,y\in X{\text{ and }}z\in .}
9572:
7001:
5829:
5171:{\textstyle \|\mathbf {x} \|_{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}}
3182:
2815:
2656:
2652:
15081:
14651:
14636:
14437:
14342:
14260:
14229:
14209:
14194:
14189:
14184:
13253:
13248:
13236:
13148:
13133:
12996:
12936:
12911:
12842:
12832:
12695:
12380:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
12242:
12097:
11785:
11773:
11333:
11192:
8946:
8253:
8107:
7163:
The generalization of the above norms to an infinite number of components leads to
6985:
6966:
5007:
4281:
4222:. Hence the formula in this case can also be written using the following notation:
4171:{\displaystyle {\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{n}\end{bmatrix}}^{\rm {T}}}
14964:
14503:
14021:
12640:
11789:
9575:
oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit
8176:
15056:
14904:
14204:
14158:
14106:
14101:
14072:
13273:
13258:
13184:
12986:
12979:
12946:
12906:
12872:
12864:
12792:
12760:
12625:
12557:
12433:
12285:
12115:
11573:
9661:
9556:
6980:
6976:
6970:
2974:
2562:{\displaystyle \|{\boldsymbol {x}}\|_{2}:={\sqrt {x_{1}^{2}+\cdots +x_{n}^{2}}}.}
2374:
197:
86:
74:
47:
14031:
11945:
10749:
7663:
Other examples of infinite-dimensional normed vector spaces can be found in the
2585:. This operation may also be referred to as "SRSS", which is an acronym for the
15087:
15036:
14751:
14393:
14245:
14046:
13843:
13791:
13451:
13317:
12964:
12954:
12573:
12525:
8744:
7698:
4403:
3431:
2958:
2344:
2067:
2030:
336:
12101:
15156:
15071:
14981:
14924:
14884:
14812:
14787:
14731:
14683:
14619:
14398:
14322:
14051:
14036:
14026:
13848:
13461:
13415:
13350:
13201:
13196:
13189:
12810:
12743:
12716:
12535:
12508:
12455:
12425:
12395:
12361:
12346:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
12334:
12253:. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag.
12187:(Third ed.). Baltimore: The Johns Hopkins University Press. p. 53.
12180:
11797:
11723:
11638:
11247:
10784:
9048:
8096:
7607:
7543:
6990:
6427:
Maximum norm (special case of: infinity norm, uniform norm, or supremum norm)
5078:
4370:
4358:{\displaystyle \|{\boldsymbol {x}}\|_{1}:=\sum _{i=1}^{n}\left|x_{i}\right|.}
4098:
4014:
2970:
2640:
114:
12316:
12268:
12127:
11647: – A topological vector space whose topology can be defined by a metric
11582: – A topological vector space whose topology can be defined by a metric
10533:{\displaystyle \|x\|_{\infty }\leq \|x\|_{2}\leq {\sqrt {n}}\|x\|_{\infty }}
725:
some authors replace property (3.) with the equivalent condition: for every
15118:
15066:
15026:
15016:
14894:
14741:
14736:
14533:
14483:
14388:
14041:
14011:
13360:
13355:
12815:
12805:
12678:
12668:
12513:
12493:
12303:
9600:
8797:
8215:
7664:
7510:
7009:
6477:
5612:
but the resulting function does not define a norm, because it violates the
4910:
4402:
The set of vectors whose 1-norm is a given constant forms the surface of a
1040:" to be a synonym of "non-negative"; these definitions are not equivalent.
174:
162:
66:
43:
12220:
11995:
11596: – All infinite-dimensional, separable Banach spaces are homeomorphic
30:
This article is about the concept in normed spaces. For field theory, see
15076:
15061:
14954:
14848:
14843:
14828:
14807:
14771:
14678:
14498:
14317:
14307:
14214:
14016:
13649:
13565:
13471:
13456:
13436:
13410:
13375:
12926:
12889:
12562:
11620:
11588:
11419:
11142:
10200:{\displaystyle C\|x\|_{\alpha }\leq \|x\|_{\beta }\leq D\|x\|_{\alpha }.}
9568:
9532:
8638:
4366:
4010:
3231:
2648:
2118:
2063:
1812:
962:
110:
54:
9518:{\displaystyle \left|\langle x,y\rangle \right|\leq \|x\|_{2}\|y\|_{2}.}
4979:{\displaystyle \|\mathbf {x} \|_{\infty }:=\max _{i}\left|x_{i}\right|.}
2478:{\displaystyle {\boldsymbol {x}}=\left(x_{1},x_{2},\ldots ,x_{n}\right)}
14889:
14802:
14766:
14626:
14508:
14250:
14082:
13405:
13386: ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (H
13370:
13211:
12959:
12721:
11593:
11579:
9604:
8227:
8103:
7005:
3460:
3285:
3217:
2659:. Hence, the Euclidean norm can be written in a coordinate-free way as
1036:" to be a synonym of "positive definite", some authors instead define "
35:
31:
12462:
8519:{\textstyle \left(\prod _{j}{\sigma _{k}(\alpha )}\right)^{p^{\mu }}.}
7025:
with quotation marks. Following Donoho's notation, the zero "norm" of
5203:
This inner product can be expressed in terms of the norm by using the
4365:
The name relates to the distance a taxi has to drive in a rectangular
15041:
14858:
13481:
12726:
12690:
11464:: any locally convex and locally bounded topological vector space is
11395:
9579:; while for the infinity norm, it is an axis-aligned square. For any
9275:
6705:
4374:
12201:
9215:{\displaystyle p(x\pm y)\geq |p(x)-p(y)|{\text{ for all }}x,y\in X.}
8592:-th root of the norm (assuming that concept makes sense) is a norm.
8177:§ Classification of seminorms: absolutely convex absorbing sets
4473:
The 1-norm is simply the sum of the absolute values of the columns.
15006:
15001:
14959:
14939:
14909:
14700:
13747:
13631:
13557:
13517:
13420:
13243:
11874:
11662:
11644:
11465:
10755:
10609:{\displaystyle \|x\|_{\infty }\leq \|x\|_{1}\leq n\|x\|_{\infty },}
10376:{\displaystyle \|x\|_{p}\leq \|x\|_{r}\leq n^{(1/r-1/p)}\|x\|_{p}.}
9666:
9657:
9540:
9302:
9254:
is a continuous linear map between normed spaces, then the norm of
7164:
7148:
4536:
3739:
3652:{\displaystyle q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} }
3313:
3221:
2921:
2785:
820:
121:
10964:{\displaystyle p_{A}(x):=\inf\{r\in \mathbb {R} :r>0,x\in rA\}}
6431:
128:. In a similar manner, a vector space with a seminorm is called a
14949:
13552:
12635:
10996:
6746:
2575:, which gives the ordinary distance from the origin to the point
1644:
is usually denoted by enclosing it within double vertical lines:
12144:(Revised 3rd ed.). New York: Springer Verlag. p. 284.
10456:{\displaystyle \|x\|_{2}\leq \|x\|_{1}\leq {\sqrt {n}}\|x\|_{2}}
9576:
9527:
5678:
class is a vector space, and it is also true that the function
3340:
where the dimensions of these spaces over the real numbers are
1530:
are equivalent if and only if they induce the same topology on
11491:
is an absolutely convex bounded neighbourhood of 0, the gauge
10750:
Classification of seminorms: absolutely convex absorbing sets
6843:{\textstyle (x_{n})\mapsto \sum _{n}{2^{-n}x_{n}/(1+x_{n})}.}
6681:
The set of vectors whose infinity norm is a given constant,
6208:
is used for absolute value of each component of the vector.
2604:
The
Euclidean norm is by far the most commonly used norm on
13983:
3776:
2920:
whose
Euclidean norm is a given positive constant forms an
1452:{\displaystyle {\tfrac {1}{C}}p\leq q\leq {\tfrac {1}{c}}p}
8796:
The characteristic feature of composition algebras is the
12158:
11920:
11849:
11847:
6960:
6573:{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n}),}
5753:{\displaystyle \int _{X}|f(x)-g(x)|^{p}~\mathrm {d} \mu }
5651:
even in the measurable analog, is that the corresponding
12213:
1804:{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}
11658:
Pages displaying short descriptions of redirect targets
11649:
Pages displaying short descriptions of redirect targets
11598:
Pages displaying short descriptions of redirect targets
11584:
Pages displaying short descriptions of redirect targets
7794:
can be constructed by combining the above; for example
3037:
as a vector in the
Euclidean plane, makes the quantity
1750:
Every (real or complex) vector space admits a norm: If
12094:
Functional analysis and control theory: Linear systems
11844:
8457:
7616:
6755:
5123:
4113:
3043:
1435:
1411:
13872:
13798:
13760:
13721:
13669:
13572:
13523:
12056:
12024:
11524:
11497:
11477:
11428:
11404:
11362:
11342:
11312:
11289:
11263:
11201:
11195:: the collection of all finite intersections of sets
11177:
11151:
11005:
10977:
10891:
10868:
10835:
10812:
10792:
10766:
10622:
10545:
10468:
10391:
10277:
10245:
10213:
10133:
10108:
10088:
10068:
10040:
10007:
9974:
9855:
9817:
9784:
9735:
9712:
9675:
9656:
In terms of the vector space, the seminorm defines a
9639:
9613:
9585:
9451:
9340:
9308:
9284:
9260:
9228:
9130:
9103:
9069:
9027:
9002:
8978:
8954:
8889:
8869:
8849:
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8753:
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8694:
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8362:
8335:
8315:
8285:
8262:
8238:
8188:
8152:
8123:
8102:
In 3D, this is similar but different for the 1-norm (
8081:
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8006:
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7946:
7800:
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7734:
7707:
7676:
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7552:
7519:
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7258:
7226:
7197:
7170:
7126:
7094:
7074:
7051:
7031:
6909:
6886:
6860:
6714:
6687:
6586:
6510:
6488:
6440:
6343:
6246:
6217:
6184:
6160:
6027:
6004:
5861:
5841:
5786:
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5684:
5657:
5622:
5583:
5477:
5432:
5387:
5251:
5213:
5184:
5086:
5048:
5019:
4992:
4923:
4895:
4875:
4855:
4825:
4792:
4680:
4620:
4593:
4573:
4547:
4524:
is not a norm because it may yield negative results.
4482:
4447:
4413:
4383:
4291:
4228:
4184:
4106:
4081:
4023:
3842:
3810:
3787:
3748:
3712:
3690:
3665:
3603:
3469:
3443:
3414:
3392:
3346:
3321:
3293:
3265:
3240:
3191:
3158:
3122:
3087:
3014:
2982:
2927:
2891:
2859:
2829:
2791:
2753:
2723:
2665:
2610:
2491:
2414:
2382:
2359:
2318:
2290:
2233:
2208:
2186:
2164:
2127:
2103:
2083:
2038:
2004:
1953:
1933:
1906:
1841:
1821:
1756:
1718:
1694:
1650:
1624:
1601:
1567:
1536:
1516:
1496:
1473:
1409:
1371:
1343:
1323:
1255:
1227:
1201:
1181:
1161:
1137:
1117:
1094:
1074:
1054:
1006:
971:
942:
907:
887:
849:
829:
794:
759:
731:
693:
665:
630:
601:
567:
541:
481:
441:
376:
345:
311:
277:
254:
225:
205:
182:
145:
81:, and is zero only at the origin. In particular, the
14467:
12408:
Topological Vector Spaces, Distributions and
Kernels
11893:
11891:
11889:
11667:
Pages displaying wikidata descriptions as a fallback
11656: – Mathematical space with a notion of distance
11617: – Property determining comparison and ordering
7509:
respectively, which can be further generalized (see
4436:. The distance derived from this norm is called the
2946:
4816:
14313:Spectral theory of ordinary differential equations
13900:
13824:
13773:
13734:
13697:
13607:
13541:
12074:
12042:
11629: – Mathematical metric in normed vector space
11555:
11510:
11483:
11447:
11410:
11386:
11348:
11324:
11298:
11275:
11227:
11183:
11163:
11126:
10987:
10963:
10877:
10848:
10821:
10798:
10772:
10730:
10608:
10532:
10455:
10375:
10263:
10231:
10199:
10120:
10094:
10074:
10046:
10026:
9993:
9958:
9841:
9799:
9770:
9721:
9694:
9645:
9625:
9591:
9517:
9433:
9321:
9290:
9266:
9246:
9214:
9112:
9089:
9035:
9013:
8989:
8965:
8937:
8875:
8855:
8835:
8812:
8784:
8735:
8703:
8677:
8621:
8584:
8551:
8518:
8443:
8413:
8368:
8344:
8321:
8301:
8268:
8244:
8203:
8167:
8138:
8087:
8067:
8047:
8015:
7992:
7964:
7932:
7786:
7750:
7720:
7689:
7654:{\textstyle \|x\|:={\sqrt {\langle x,x\rangle }}.}
7653:
7596:
7565:
7534:
7501:
7464:
7242:
7210:
7183:
7139:
7100:
7080:
7060:
7037:
6945:
6895:
6872:
6842:
6726:
6696:
6671:
6572:
6496:
6465:
6416:
6329:
6232:
6200:
6166:
6146:
6013:
5988:
5847:
5808:
5772:
5752:
5670:
5643:
5604:
5567:
5459:
5415:
5373:
5229:
5195:
5170:
5110:{\displaystyle \langle \,\cdot ,\,\cdot \rangle ,}
5109:
5069:
5034:
4998:
4978:
4901:
4881:
4861:
4837:
4807:
4778:
4667:{\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})}
4666:
4606:
4579:
4559:
4516:
4460:
4426:
4392:
4357:
4265:
4199:
4170:
4089:
4067:
3999:
3828:
3793:
3766:
3730:
3698:
3676:
3651:
3589:
3451:
3422:
3400:
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3304:
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3173:
3144:
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2702:
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2194:
2172:
2150:
2109:
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2013:
1990:
1939:
1919:
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1734:
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1021:
992:
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928:
893:
869:
835:
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616:
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553:
527:
462:
427:
351:
327:
297:
260:
236:
211:
188:
153:
12341:
12207:
11886:
11880:
11641: – Measure of the "size" of linear operators
7443:
7385:
7331:
7280:
4275:
15154:
12050:where it coincides with the Euclidean norm, and
11774:"Pseudo-normed linear spaces and Abelian groups"
10979:
10914:
7870:
7045:is simply the number of non-zero coordinates of
6608:
4946:
1991:{\displaystyle \sum _{i\in I}\left|s_{i}\right|}
12284:. Vol. 936. Berlin, Heidelberg, New York:
12085:
11576: – Generalization of the concept of a norm
8883:of the composition algebra, its norm satisfies
8414:{\displaystyle \left\{\sigma _{j}\right\}_{j},}
7513:). These norms are also valid in the limit as
4009:In this case, the norm can be expressed as the
1893:{\displaystyle x=\sum _{i\in I}s_{i}x_{i}\in X}
1742:with single vertical lines is also widespread.
1088:are two norms (or seminorms) on a vector space
901:is a norm (or more generally, a seminorm) then
109:of the vector. This norm can be defined as the
13825:{\displaystyle S\left(\mathbb {R} ^{n}\right)}
12440:. Mineola, New York: Dover Publications, Inc.
12342:Narici, Lawrence; Beckenstein, Edward (2011).
11831:
8099:of a particular shape, size, and orientation.
7474:for complex-valued sequences and functions on
5577:This definition is still of some interest for
14453:
13969:
13933:Mathematical formulation of quantum mechanics
12478:
12368:
12179:
2408:the intuitive notion of length of the vector
2062:is a norm on the vector space formed by the
1155:, if there exist two positive real constants
12278:Counterexamples in Topological Vector Spaces
11550:
11531:
11381:
11369:
11222:
11202:
10958:
10917:
10716:
10709:
10694:
10687:
10668:
10661:
10649:
10642:
10630:
10623:
10594:
10587:
10572:
10565:
10553:
10546:
10521:
10514:
10495:
10488:
10476:
10469:
10444:
10437:
10418:
10411:
10399:
10392:
10361:
10354:
10304:
10297:
10285:
10278:
10185:
10178:
10163:
10156:
10144:
10137:
10015:
10008:
9982:
9975:
9904:
9892:
9886:
9874:
9868:
9856:
9833:
9827:
9689:
9676:
9503:
9496:
9487:
9480:
9469:
9457:
9392:
9385:
9376:
9369:
9358:
9346:
8039:
8030:
7807:
7801:
7643:
7631:
7623:
7617:
7365:
7358:
7266:
7259:
7108:approaches 0. Of course, the zero "norm" is
6916:
6910:
6867:
6861:
6596:
6587:
6448:
6441:
6399:
6390:
6370:
6361:
6312:
6303:
6278:
6269:
6118:
6109:
6043:
6034:
5960:
5951:
5893:
5884:
5491:
5478:
5309:
5252:
5163:
5147:
5133:
5124:
5101:
5087:
5058:
5049:
4933:
4924:
4690:
4681:
4301:
4292:
4237:
4229:
4032:
4024:
3851:
3843:
3476:
3470:
3211:
2674:
2666:
2501:
2492:
1900:(where all but finitely many of the scalars
1657:
1651:
12438:Modern Methods in Topological Vector Spaces
12275:
11724:"Pseudo-norm - Encyclopedia of Mathematics"
11694:
10829:To each such subset corresponds a seminorm
9771:{\displaystyle \left\|v_{n}-v\right\|\to 0}
8637:share the usual properties of a norm since
8182:All the above formulas also yield norms on
7502:{\displaystyle X\subseteq \mathbb {R} ^{n}}
46:. For norms in descriptive set theory, see
14460:
14446:
13976:
13962:
12485:
12471:
11810:
11674: – Type of function in linear algebra
11623: – Norm on a vector space of matrices
8526:As that function is homogeneous of degree
4141:
4137:
4126:
3684:This is the same as the Euclidean norm on
3008:This identification of the complex number
13808:
13608:{\displaystyle B_{p,q}^{s}(\mathbb {R} )}
13598:
12059:
12027:
11865:
10927:
10248:
9083:
9029:
9004:
8980:
8956:
8191:
8155:
8126:
7949:
7774:
7744:
7590:
7489:
7297:
7236:
5553:
5097:
5090:
5056:
5052:
3813:
3751:
3715:
3692:
3667:
3643:
3631:
3619:
3532:
3508:
3484:
3445:
3416:
3394:
3323:
3295:
3267:
3242:
3195:
2985:
2894:
2613:
2385:
2298:
2210:
2188:
2166:
2135:
1581:
1310:{\displaystyle cq(x)\leq p(x)\leq Cq(x).}
863:
291:
227:
150:
146:
14266:Group algebra of a locally compact group
12432:
12310:
12241:
12175:
12173:
12091:
11926:
11853:
9526:
8117:of a centrally-symmetric convex body in
6430:
5820:. These spaces are of great interest in
3777:Finite-dimensional complex normed spaces
1688:Such notation is also sometimes used if
13698:{\displaystyle L^{\lambda ,p}(\Omega )}
12492:
12247:Topological Vector Spaces: Chapters 1–5
11690:
11688:
11237:locally convex topological vector space
11139:locally convex topological vector space
8595:
6946:{\displaystyle \lVert x\rVert =d(x,0).}
5780:th root) defines a distance that makes
4296:
4254:
4246:
4233:
4187:
4083:
4056:
4042:
4028:
3847:
2691:
2683:
2670:
2496:
2416:
89:is defined by a norm on the associated
14:
15155:
14599:Uniform boundedness (Banach–Steinhaus)
13938:Ordinary Differential Equations (ODEs)
13052:Banach–Steinhaus (Uniform boundedness)
12402:
12164:
11968:
11605: – Periodicity computation method
8221:
7158:
6961:Hamming distance of a vector from zero
4377:) to get from the origin to the point
3379:{\displaystyle 1,2,4,{\text{ and }}8,}
2712:The Euclidean norm is also called the
2024:
14441:
13957:
12466:
12410:. Mineola, N.Y.: Dover Publications.
12170:
11993:
11943:
11939:
11937:
11935:
11771:
8146:(centered at zero) defines a norm on
7535:{\displaystyle p\rightarrow +\infty }
5077:-norm is even induced by a canonical
4200:{\displaystyle {\boldsymbol {x}}^{H}}
3738:Similarly, the canonical norm on the
3386:respectively. The canonical norms on
3145:{\displaystyle {\sqrt {{\bar {z}}z}}}
2284:This isomorphism is given by sending
305:with the following properties, where
12139:
11685:
11611: – Statistical distance measure
9994:{\displaystyle \|\cdot \|_{\alpha }}
6873:{\displaystyle \lVert \cdot \rVert }
4517:{\displaystyle \sum _{i=1}^{n}x_{i}}
3434:functions, as discussed previously.
2151:{\displaystyle f:\mathbb {F} \to X,}
1835:then the real-valued map that sends
428:{\displaystyle p(x+y)\leq p(x)+p(y)}
10027:{\displaystyle \|\cdot \|_{\beta }}
9090:{\displaystyle p:X\to \mathbb {R} }
8641:are allowed. A composition algebra
7147:norm, echoing the notation for the
4846:
1709:
1588:{\displaystyle p:X\to \mathbb {R} }
1043:
870:{\displaystyle p:X\to \mathbb {R} }
298:{\displaystyle p:X\to \mathbb {R} }
94:
24:
13766:
13727:
13689:
13533:
11932:
11897:
11834:Quantum Mechanics in Hilbert Space
10720:
10634:
10598:
10557:
10525:
10480:
9791:
7760:
7610:induces in a natural way the norm
7585:
7558:
7529:
7433:
6600:
6452:
6350:
6346:
6253:
6249:
6054:
6031:
5868:
5864:
5743:
5555:
5442:
5070:{\displaystyle \|\,\cdot \,\|_{2}}
4937:
4876:
4162:
3116:, the norm can also be written as
3074:{\textstyle {\sqrt {x^{2}+y^{2}}}}
2953:Dot product § Complex vectors
2913:{\displaystyle \mathbb {R} ^{n+1}}
2097:on a one-dimensional vector space
77:with scaling, obeys a form of the
25:
15184:
13430:Subsets / set operations
13207:Differentiation in Fréchet spaces
12075:{\displaystyle \mathbb {R} ^{0},}
12043:{\displaystyle \mathbb {R} ^{1},}
10264:{\displaystyle \mathbb {C} ^{n},}
9555:. In particular, every norm is a
7965:{\displaystyle \mathbb {R} ^{4}.}
6466:{\displaystyle \|x\|_{\infty }=1}
5460:{\displaystyle (X,\Sigma ,\mu ),}
4090:{\displaystyle {\boldsymbol {x}}}
3829:{\displaystyle \mathbb {C} ^{n},}
3767:{\displaystyle \mathbb {R} ^{8}.}
3731:{\displaystyle \mathbb {R} ^{4}.}
3001:{\displaystyle \mathbb {R} ^{2}.}
2947:Euclidean norm of complex numbers
2629:{\displaystyle \mathbb {R} ^{n},}
2401:{\displaystyle \mathbb {R} ^{n},}
2338:
2305:{\displaystyle 1\in \mathbb {F} }
956:also has the following property:
15137:
15136:
14422:
14421:
14348:Topological quantum field theory
11635: – Function made from a set
11556:{\displaystyle X=\{g_{X}<1\}}
11246:Such a method is used to design
10760:All seminorms on a vector space
8204:{\displaystyle \mathbb {C} ^{n}}
8168:{\displaystyle \mathbb {R} ^{n}}
8139:{\displaystyle \mathbb {R} ^{n}}
7787:{\displaystyle \mathbb {R} ^{n}}
7112:truly a norm, because it is not
6591:
6512:
6490:
6394:
6385:
6365:
6354:
6307:
6273:
6113:
6085:
6072:
6058:
6038:
5955:
5888:
5616:. What is true for this case of
5186:
5159:
5151:
5128:
4928:
4685:
4622:
3645:
3633:
3621:
15124:With the approximation property
13735:{\displaystyle \ell ^{\infty }}
12313:The Elements of Operator Theory
12235:
12183:; Van Loan, Charles F. (1996).
12133:
12012:
11987:
11962:
11603:Least-squares spectral analysis
9547:. In turn, every seminorm is a
9401:
8938:{\displaystyle N(wz)=N(w)N(z).}
7566:{\displaystyle \ell ^{\infty }}
5998:The derivative with respect to
3706:considered as the vector space
2227:and norm-preserving means that
1396:{\displaystyle cq\leq p\leq Cq}
38:. For commutative algebra, see
14587:Open mapping (Banach–Schauder)
13895:
13876:
13692:
13686:
13602:
13594:
13536:
13530:
13124:Lomonosov's invariant subspace
13047:Banach–Schauder (open mapping)
11971:Dynamics of Structures, 4th Ed
11859:
11804:
11765:
11740:
11716:
11319:
11313:
11270:
11264:
11158:
11152:
11107:
11101:
11039:
11033:
10908:
10902:
10780:can be classified in terms of
10349:
10321:
9950:
9938:
9842:{\displaystyle (X,\|\cdot \|)}
9836:
9818:
9788:
9762:
9758:
9737:
9562:
9441:A special case of this is the
9362:
9342:
9238:
9185:
9181:
9175:
9166:
9160:
9153:
9146:
9134:
9079:
8929:
8923:
8917:
8911:
8902:
8893:
8763:
8757:
8672:
8648:
8616:
8610:
8579:
8567:
8546:
8534:
8491:
8485:
7919:
7873:
7597:{\displaystyle L^{\infty }\,.}
7523:
7419:
7414:
7408:
7401:
6937:
6925:
6833:
6814:
6772:
6769:
6756:
6564:
6519:
6194:
6186:
6091:
6080:
5835:The partial derivative of the
5803:
5797:
5729:
5724:
5718:
5709:
5703:
5696:
5550:
5544:
5532:
5526:
5451:
5433:
5410:
5398:
4661:
4629:
4276:Taxicab norm or Manhattan norm
4214:This formula is valid for any
3980:
3942:
3742:is just the Euclidean norm on
3165:
3131:
2268:
2265:
2259:
2253:
2243:
2235:
2139:
2048:
2040:
1728:
1720:
1672:
1666:
1577:
1301:
1295:
1283:
1277:
1268:
1262:
981:
975:
917:
911:
859:
769:
763:
703:
697:
687:Because property (2.) implies
640:
634:
522:
516:
509:
501:
494:
485:
422:
416:
407:
401:
392:
380:
321:
313:
287:
13:
1:
14144:Uniform boundedness principle
12208:Narici & Beckenstein 2011
11881:Narici & Beckenstein 2011
11790:10.1215/s0012-7094-39-00551-x
11678:
11654:Relation of norms and metrics
9800:{\displaystyle n\to \infty .}
9058:
9014:{\displaystyle \mathbb {H} ,}
8990:{\displaystyle \mathbb {C} ,}
8966:{\displaystyle \mathbb {R} ,}
7701:than an infinite-dimensional
5416:{\displaystyle L^{2}(X,\mu )}
5196:{\displaystyle \mathbf {x} .}
3677:{\displaystyle \mathbb {H} .}
3333:{\displaystyle \mathbb {O} ,}
3305:{\displaystyle \mathbb {H} ,}
3277:{\displaystyle \mathbb {C} ,}
3252:{\displaystyle \mathbb {R} ,}
3234:. These are the real numbers
2220:{\displaystyle \mathbb {C} ,}
2070:. The complex numbers form a
528:{\displaystyle p(sx)=|s|p(x)}
237:{\displaystyle \mathbb {C} ,}
168:
13009:Singular value decomposition
12372:; Wolff, Manfred P. (1999).
12311:Kubrusly, Carlos S. (2011).
12282:Lecture Notes in Mathematics
11387:{\displaystyle A=\{p<1\}}
11228:{\displaystyle \{p<1/n\}}
10232:{\displaystyle p>r\geq 1}
9660:on the space, and this is a
9036:{\displaystyle \mathbb {O} }
8678:{\displaystyle (A,{}^{*},N)}
8444:{\displaystyle \alpha \in E}
8000:we can define a new norm of
6880:on an F-space with distance
6736:
6497:{\displaystyle \mathbf {x} }
5644:{\displaystyle 0<p<1,}
5605:{\displaystyle 0<p<1,}
5536:
5356:
3699:{\displaystyle \mathbb {H} }
3452:{\displaystyle \mathbb {H} }
3423:{\displaystyle \mathbb {C} }
3401:{\displaystyle \mathbb {R} }
2277:{\displaystyle |x|=p(f(x)).}
2195:{\displaystyle \mathbb {R} }
2173:{\displaystyle \mathbb {F} }
2072:one-dimensional vector space
7:
14808:Radially convex/Star-shaped
14793:Pre-compact/Totally bounded
13774:{\displaystyle L^{\infty }}
13542:{\displaystyle ba(\Sigma )}
13411:Radially convex/Star-shaped
12315:(Second ed.). Boston:
12276:Khaleelulla, S. M. (1982).
11772:Hyers, D. H. (1939-09-01).
11567:
9122:reverse triangle inequality
8785:{\displaystyle N(z)=zz^{*}}
5469:square-integrable functions
2485:is captured by the formula
1745:
1681:{\displaystyle \|z\|=p(z).}
1595:is given on a vector space
1556:
1467:on the set of all norms on
1221:such that for every vector
98:
10:
15189:
14494:Continuous linear operator
14287:Invariant subspace problem
13901:{\displaystyle W(X,L^{p})}
12225:Mathematics Stack Exchange
12221:"Relation between p-norms"
11813:Real Mathematical Analysis
10753:
8225:
8110:with parallelogram base).
7243:{\displaystyle p\geq 1\,,}
6964:
6854:some real-valued function
6475:
5237:this inner product is the
5230:{\displaystyle \ell ^{2},}
4534:
4279:
4017:of the vector and itself:
3228:Euclidean Hurwitz algebras
3215:
3174:{\displaystyle {\bar {z}}}
2950:
2342:
1618:then the norm of a vector
993:{\displaystyle p(x)\geq 0}
29:
15132:
14877:
14839:Algebraic interior (core)
14821:
14719:
14607:
14581:Vector-valued Hahn–Banach
14542:
14476:
14469:Topological vector spaces
14417:
14376:
14300:
14279:
14238:
14177:
14119:
14065:
14007:
14000:
13915:
13500:
13447:Algebraic interior (core)
13429:
13338:
13172:
13062:Cauchy–Schwarz inequality
13017:
12945:
12791:
12705:Function space Topologies
12704:
12618:
12501:
12374:Topological Vector Spaces
12344:Topological Vector Spaces
12102:10.1007/978-94-015-7758-8
12092:Rolewicz, Stefan (1987),
11778:Duke Mathematical Journal
11248:weak and weak* topologies
10999:, with the property that
9695:{\displaystyle \{v_{n}\}}
9443:Cauchy–Schwarz inequality
8302:{\displaystyle p^{\mu },}
7751:{\displaystyle p<q\,.}
7721:{\displaystyle \ell ^{q}}
7690:{\displaystyle \ell ^{p}}
7184:{\displaystyle \ell ^{p}}
6749:of sequences with F–norm
6504:is some vector such that
4607:{\displaystyle \ell ^{p}}
4527:
4461:{\displaystyle \ell ^{1}}
4427:{\displaystyle \ell ^{1}}
3212:Quaternions and octonions
2873:{\displaystyle \ell ^{2}}
2767:{\displaystyle \ell ^{2}}
463:{\displaystyle x,y\in X.}
154:{\displaystyle \,\leq \,}
117:of a vector with itself.
14669:Topological homomorphism
14529:Topological vector space
14256:Spectrum of a C*-algebra
12251:Éléments de mathématique
11422:neighbourhood of 0, and
11418:is an absolutely convex
10988:{\displaystyle \inf _{}}
9553:properties of the latter
9545:properties of the latter
9247:{\displaystyle u:X\to Y}
9053:isotropic quadratic form
8106:) and the maximum norm (
6989:, which is important in
6211:For the special case of
6201:{\displaystyle |\cdot |}
5818:topological vector space
5809:{\displaystyle L^{p}(X)}
5471:, this inner product is
5006:-norm is related to the
3836:the most common norm is
2957:The Euclidean norm of a
219:of the complex numbers
42:. For group theory, see
40:Absolute value (algebra)
27:Length in a vector space
14353:Noncommutative geometry
11832:PrugoveÄŤki, E. (1981).
11615:Magnitude (mathematics)
11460:The converse is due to
11448:{\displaystyle p=p_{A}}
11235:turns the space into a
9849:is a normed space then
9626:{\displaystyle p\geq 1}
9551:and thus satisfies all
9543:and thus satisfies all
9045:definite quadratic form
8736:{\displaystyle {}^{*},}
8329:have algebraic closure
8048:{\displaystyle \|Ax\|.}
7699:strictly finer topology
6704:forms the surface of a
5816:into a complete metric
5241:Euclidean inner product
4882:{\displaystyle \infty }
4560:{\displaystyle p\geq 1}
3226:There are exactly four
2973:is identified with the
1243:{\displaystyle x\in X,}
1022:{\displaystyle x\in X.}
961:
747:{\displaystyle x\in X,}
718:{\displaystyle p(0)=0,}
617:{\displaystyle x\in X,}
130:seminormed vector space
65:from a real or complex
14727:Absolutely convex/disk
14409:Tomita–Takesaki theory
14384:Approximation property
14328:Calculus of variations
13902:
13826:
13775:
13736:
13699:
13609:
13543:
12712:Banach–Mazur compactum
12502:Types of Banach spaces
12076:
12044:
11907:kconrad.math.uconn.edu
11900:"Equivalence of norms"
11728:encyclopediaofmath.org
11699:. Birkhäuser. p.
11557:
11512:
11485:
11449:
11412:
11388:
11350:
11326:
11300:
11277:
11229:
11185:
11165:
11128:
10989:
10965:
10879:
10850:
10823:
10800:
10774:
10732:
10610:
10534:
10457:
10377:
10265:
10233:
10201:
10122:
10121:{\displaystyle x\in X}
10096:
10076:
10048:
10028:
9995:
9960:
9843:
9801:
9772:
9723:
9696:
9647:
9627:
9593:
9536:
9519:
9435:
9323:
9292:
9268:
9248:
9216:
9114:
9091:
9037:
9015:
8991:
8967:
8939:
8877:
8857:
8837:
8814:
8786:
8737:
8705:
8679:
8623:
8586:
8553:
8520:
8445:
8415:
8370:
8346:
8323:
8303:
8270:
8246:
8211:without modification.
8205:
8169:
8140:
8089:
8069:
8049:
8017:
7994:
7966:
7934:
7788:
7752:
7722:
7691:
7655:
7598:
7567:
7536:
7503:
7466:
7244:
7212:
7185:
7141:
7102:
7082:
7062:
7039:
6947:
6897:
6874:
6844:
6728:
6698:
6673:
6574:
6498:
6473:
6467:
6418:
6331:
6234:
6202:
6168:
6167:{\displaystyle \circ }
6148:
6015:
5990:
5849:
5810:
5774:
5754:
5672:
5645:
5606:
5569:
5467:which consists of all
5461:
5417:
5375:
5231:
5197:
5172:
5111:
5071:
5036:
5000:
4980:
4903:
4883:
4863:
4839:
4809:
4780:
4728:
4668:
4608:
4581:
4567:be a real number. The
4561:
4518:
4503:
4462:
4428:
4394:
4359:
4333:
4267:
4201:
4172:
4091:
4069:
4001:
3830:
3795:
3768:
3732:
3700:
3678:
3653:
3591:
3453:
3437:The canonical norm on
3424:
3402:
3380:
3334:
3306:
3278:
3253:
3203:
3175:
3146:
3110:
3109:{\displaystyle z=x+iy}
3075:
3031:
3002:
2935:
2914:
2885:The set of vectors in
2874:
2844:
2806:
2768:
2738:
2704:
2645:Euclidean vector space
2630:
2581:—a consequence of the
2563:
2479:
2402:
2367:
2329:
2306:
2278:
2221:
2196:
2174:
2152:
2111:
2091:
2056:
2015:
1992:
1941:
1921:
1894:
1829:
1805:
1736:
1702:
1682:
1638:
1637:{\displaystyle z\in X}
1612:
1589:
1547:
1524:
1504:
1484:
1453:
1397:
1351:
1331:
1311:
1244:
1215:
1214:{\displaystyle c>0}
1189:
1169:
1145:
1125:
1105:
1082:
1062:
1023:
994:
950:
930:
929:{\displaystyle p(0)=0}
895:
871:
837:
808:
782:
781:{\displaystyle p(x)=0}
748:
719:
679:
653:
652:{\displaystyle p(x)=0}
618:
578:
555:
554:{\displaystyle x\in X}
529:
464:
429:
353:
329:
299:
262:
238:
213:
190:
155:
91:Euclidean vector space
14762:Complemented subspace
14576:hyperplane separation
14404:Banach–Mazur distance
14367:Generalized functions
13928:Finite element method
13923:Differential operator
13903:
13827:
13776:
13737:
13700:
13610:
13544:
13384:Convex series related
13180:Abstract Wiener space
13107:hyperplane separation
12662:Minkowski functionals
12546:Polarization identity
12140:Lang, Serge (2002) .
12077:
12045:
12000:mathworld.wolfram.com
11969:Chopra, Anil (2012).
11950:mathworld.wolfram.com
11558:
11513:
11511:{\displaystyle g_{X}}
11486:
11450:
11413:
11389:
11351:
11327:
11301:
11278:
11230:
11186:
11166:
11129:
10990:
10966:
10880:
10851:
10849:{\displaystyle p_{A}}
10824:
10801:
10775:
10733:
10611:
10535:
10458:
10378:
10266:
10234:
10202:
10123:
10097:
10077:
10049:
10029:
9996:
9961:
9844:
9802:
9773:
9724:
9697:
9648:
9628:
9594:
9530:
9520:
9436:
9324:
9322:{\displaystyle L^{p}}
9293:
9269:
9249:
9217:
9115:
9092:
9038:
9016:
8992:
8968:
8940:
8878:
8858:
8838:
8815:
8787:
8738:
8706:
8680:
8624:
8587:
8554:
8521:
8446:
8423:Galois-theoretic norm
8416:
8371:
8347:
8324:
8304:
8271:
8247:
8206:
8170:
8141:
8090:
8070:
8050:
8018:
7995:
7979:linear transformation
7974:For any norm and any
7967:
7935:
7789:
7753:
7723:
7692:
7656:
7599:
7568:
7537:
7504:
7467:
7245:
7213:
7211:{\displaystyle L^{p}}
7186:
7142:
7140:{\displaystyle L^{0}}
7103:
7083:
7063:
7040:
6948:
6898:
6875:
6845:
6729:
6699:
6674:
6575:
6499:
6468:
6434:
6419:
6332:
6235:
6203:
6169:
6149:
6016:
5991:
5850:
5811:
5775:
5755:
5673:
5671:{\displaystyle L^{p}}
5646:
5607:
5570:
5462:
5418:
5376:
5232:
5205:polarization identity
5198:
5173:
5112:
5072:
5037:
5001:
4981:
4909:-norm approaches the
4904:
4884:
4864:
4840:
4810:
4781:
4708:
4669:
4609:
4582:
4562:
4519:
4483:
4463:
4429:
4395:
4360:
4313:
4268:
4202:
4173:
4092:
4070:
4002:
3831:
3796:
3769:
3733:
3701:
3679:
3654:
3597:for every quaternion
3592:
3454:
3425:
3403:
3381:
3335:
3307:
3279:
3254:
3204:
3176:
3147:
3111:
3076:
3032:
3003:
2936:
2915:
2875:
2845:
2843:{\displaystyle L^{2}}
2807:
2805:{\displaystyle L^{p}}
2769:
2739:
2737:{\displaystyle L^{2}}
2705:
2631:
2564:
2480:
2403:
2368:
2343:Further information:
2330:
2307:
2279:
2222:
2197:
2175:
2153:
2112:
2092:
2057:
2016:
1993:
1942:
1922:
1920:{\displaystyle s_{i}}
1895:
1830:
1806:
1737:
1703:
1683:
1639:
1613:
1590:
1548:
1525:
1505:
1485:
1454:
1398:
1352:
1332:
1312:
1245:
1216:
1190:
1170:
1146:
1126:
1106:
1083:
1063:
1024:
995:
951:
931:
896:
872:
838:
809:
783:
749:
720:
680:
654:
619:
587:Positive definiteness
579:
556:
530:
465:
430:
354:
330:
300:
263:
239:
214:
191:
156:
101:, or, sometimes, the
15012:Locally convex space
14562:Closed graph theorem
14514:Locally convex space
14149:Kakutani fixed-point
14134:Riesz representation
13870:
13796:
13758:
13719:
13667:
13570:
13521:
13510:Absolute continuity
13164:Schauder fixed-point
13154:Riesz representation
13114:Kakutani fixed-point
13082:Freudenthal spectral
12568:L-semi-inner product
12082:where it is trivial.
12054:
12022:
11815:. Springer. p.
11695:Knapp, A.W. (2005).
11633:Minkowski functional
11609:Mahalanobis distance
11522:
11495:
11475:
11426:
11402:
11360:
11340:
11310:
11287:
11261:
11199:
11175:
11149:
11003:
10975:
10889:
10866:
10833:
10810:
10790:
10764:
10744:uniformly isomorphic
10620:
10543:
10466:
10389:
10275:
10243:
10211:
10131:
10106:
10086:
10066:
10038:
10005:
9972:
9853:
9815:
9782:
9733:
9710:
9673:
9637:
9611:
9583:
9449:
9338:
9306:
9282:
9274:and the norm of the
9258:
9226:
9128:
9101:
9067:
9025:
9000:
8976:
8952:
8887:
8867:
8847:
8824:
8804:
8751:
8718:
8692:
8687:algebra over a field
8645:
8631:composition algebras
8622:{\displaystyle N(z)}
8604:
8600:The concept of norm
8596:Composition algebras
8564:
8531:
8455:
8429:
8380:
8360:
8333:
8313:
8283:
8260:
8236:
8186:
8150:
8121:
8115:Minkowski functional
8079:
8059:
8027:
8004:
7984:
7944:
7798:
7769:
7732:
7705:
7674:
7614:
7577:
7550:
7517:
7478:
7256:
7224:
7195:
7168:
7153:measurable functions
7124:
7114:positive homogeneous
7092:
7072:
7049:
7029:
6907:
6884:
6858:
6753:
6743:measurable functions
6712:
6685:
6584:
6508:
6486:
6438:
6341:
6244:
6233:{\displaystyle p=2,}
6215:
6182:
6158:
6025:
6002:
5859:
5839:
5784:
5764:
5682:
5655:
5620:
5581:
5475:
5430:
5385:
5381:while for the space
5249:
5211:
5182:
5121:
5084:
5046:
5035:{\displaystyle p=2,}
5017:
4990:
4921:
4893:
4873:
4853:
4823:
4808:{\displaystyle p=1,}
4790:
4678:
4618:
4591:
4571:
4545:
4480:
4445:
4411:
4381:
4289:
4226:
4182:
4104:
4097:is represented as a
4079:
4021:
3840:
3808:
3785:
3746:
3710:
3688:
3663:
3601:
3467:
3441:
3412:
3390:
3344:
3319:
3291:
3263:
3259:the complex numbers
3238:
3202:{\displaystyle z\,.}
3189:
3156:
3120:
3085:
3041:
3030:{\displaystyle x+iy}
3012:
2980:
2925:
2889:
2857:
2827:
2789:
2751:
2721:
2663:
2643:of two vectors of a
2608:
2489:
2412:
2380:
2357:
2316:
2312:to a vector of norm
2288:
2231:
2206:
2184:
2162:
2125:
2101:
2081:
2036:
2002:
1951:
1931:
1904:
1839:
1819:
1754:
1716:
1692:
1648:
1622:
1599:
1565:
1534:
1514:
1494:
1471:
1465:equivalence relation
1463:and thus defines an
1407:
1369:
1341:
1321:
1253:
1225:
1199:
1179:
1159:
1135:
1115:
1092:
1072:
1052:
1004:
969:
940:
905:
885:
879:sublinear functional
847:
827:
807:{\displaystyle x=0.}
792:
757:
729:
691:
678:{\displaystyle x=0.}
663:
628:
599:
565:
539:
479:
473:Absolute homogeneity
439:
374:
343:
309:
275:
270:real-valued function
252:
223:
203:
180:
143:
15173:Norms (mathematics)
15163:Functional analysis
14992:Interpolation space
14524:Operator topologies
14333:Functional calculus
14292:Mahler's conjecture
14271:Von Neumann algebra
13985:Functional analysis
13593:
13331:measurable function
13281:Functional calculus
13144:Parseval's identity
13057:Bessel's inequality
13004:Polar decomposition
12783:Uniform convergence
12541:Inner product space
12370:Schaefer, Helmut H.
12210:, pp. 107–113.
12185:Matrix Computations
12167:, pp. 242–243.
11994:Weisstein, Eric W.
11944:Weisstein, Eric W.
11883:, pp. 120–121.
11868:Functional Analysis
11811:Pugh, C.C. (2015).
11697:Basic Real Analysis
11325:{\displaystyle (p)}
11276:{\displaystyle (p)}
11239:so that every p is
11164:{\displaystyle (p)}
9909: for all
9535:in different norms.
9333:Hölder's inequality
9191: for all
8793:called the "norm".
8222:In abstract algebra
7159:Infinite dimensions
7118:Abusing terminology
6727:{\displaystyle 2c.}
6137:
5979:
5822:functional analysis
5614:triangle inequality
4838:{\displaystyle p=2}
4587:-norm (also called
4220:complex dot product
4216:inner product space
4209:conjugate transpose
2583:Pythagorean theorem
2553:
2529:
2055:{\displaystyle |x|}
2025:Absolute-value norm
1815:for a vector space
1735:{\displaystyle |x|}
368:Triangle inequality
328:{\displaystyle |s|}
126:normed vector space
79:triangle inequality
15022:(Pseudo)Metrizable
14854:Minkowski addition
14706:Sublinear function
14358:Riemann hypothesis
14057:Topological vector
13943:Validated numerics
13898:
13854:Sobolev inequality
13822:
13771:
13732:
13695:
13624:Bounded variation
13605:
13573:
13558:Banach coordinate
13539:
13477:Minkowski addition
13139:M. Riesz extension
12619:Banach spaces are:
12072:
12040:
11866:Rudin, W. (1991).
11672:Sublinear function
11627:Minkowski distance
11553:
11508:
11481:
11445:
11408:
11384:
11346:
11322:
11299:{\displaystyle p:}
11296:
11283:contains a single
11273:
11225:
11181:
11161:
11124:
10985:
10984:
10961:
10878:{\displaystyle A,}
10875:
10846:
10822:{\displaystyle X.}
10819:
10796:
10770:
10728:
10606:
10530:
10453:
10373:
10261:
10229:
10197:
10118:
10102:such that for all
10092:
10072:
10044:
10034:on a vector space
10024:
9991:
9956:
9839:
9797:
9768:
9722:{\displaystyle v,}
9719:
9692:
9643:
9623:
9589:
9549:sublinear function
9537:
9515:
9431:
9319:
9288:
9264:
9244:
9212:
9113:{\displaystyle X,}
9110:
9097:on a vector space
9087:
9033:
9011:
8987:
8963:
8935:
8873:
8853:
8836:{\displaystyle wz}
8833:
8820:: for the product
8810:
8782:
8733:
8704:{\displaystyle A,}
8701:
8675:
8619:
8582:
8549:
8516:
8473:
8441:
8411:
8366:
8345:{\displaystyle K.}
8342:
8319:
8299:
8278:inseparable degree
8266:
8242:
8201:
8165:
8136:
8085:
8065:
8045:
8016:{\displaystyle x,}
8013:
7990:
7962:
7930:
7784:
7748:
7718:
7687:
7651:
7594:
7563:
7532:
7499:
7462:
7302:
7240:
7208:
7181:
7137:
7098:
7078:
7061:{\displaystyle x,}
7058:
7035:
6995:information theory
6943:
6896:{\displaystyle d,}
6893:
6870:
6840:
6784:
6724:
6697:{\displaystyle c,}
6694:
6669:
6570:
6494:
6474:
6463:
6414:
6327:
6230:
6198:
6164:
6144:
6117:
6014:{\displaystyle x,}
6011:
5986:
5959:
5855:-norm is given by
5845:
5826:probability theory
5806:
5770:
5750:
5668:
5641:
5602:
5565:
5457:
5423:associated with a
5413:
5371:
5343:
5227:
5193:
5168:
5107:
5067:
5032:
4996:
4976:
4954:
4899:
4879:
4859:
4835:
4805:
4776:
4664:
4604:
4577:
4557:
4514:
4458:
4438:Manhattan distance
4424:
4393:{\displaystyle x.}
4390:
4369:(like that of the
4355:
4263:
4197:
4168:
4154:
4087:
4065:
3997:
3826:
3791:
3764:
3728:
3696:
3674:
3649:
3587:
3449:
3420:
3398:
3376:
3330:
3302:
3274:
3249:
3199:
3171:
3142:
3106:
3071:
3027:
2998:
2931:
2910:
2870:
2840:
2802:
2764:
2734:
2700:
2653:coordinate vectors
2626:
2559:
2539:
2515:
2475:
2398:
2363:
2349:Euclidean distance
2328:{\displaystyle 1,}
2325:
2302:
2274:
2217:
2192:
2170:
2148:
2107:
2087:
2052:
2014:{\displaystyle X.}
2011:
1988:
1969:
1937:
1917:
1890:
1863:
1825:
1801:
1732:
1698:
1678:
1634:
1611:{\displaystyle X,}
1608:
1585:
1546:{\displaystyle X.}
1543:
1520:
1500:
1483:{\displaystyle X.}
1480:
1449:
1444:
1420:
1393:
1347:
1327:
1307:
1240:
1211:
1185:
1165:
1141:
1121:
1104:{\displaystyle X.}
1101:
1078:
1058:
1019:
990:
946:
926:
891:
867:
833:
804:
778:
744:
715:
675:
649:
614:
577:{\displaystyle s.}
574:
551:
525:
460:
425:
349:
335:denotes the usual
325:
295:
258:
234:
209:
186:
151:
83:Euclidean distance
34:. For ideals, see
15150:
15149:
14869:Relative interior
14615:Bilinear operator
14499:Linear functional
14435:
14434:
14338:Integral operator
14115:
14114:
13951:
13950:
13663:Morrey–Campanato
13645:compact Hausdorff
13492:Relative interior
13346:Absolutely convex
13313:Projection-valued
12922:Strictly singular
12848:on Hilbert spaces
12609:of Hilbert spaces
12447:978-0-486-49353-4
12417:978-0-486-45352-1
12387:978-1-4612-7155-0
12326:978-0-8176-4998-2
12295:978-3-540-11565-6
12243:Bourbaki, Nicolas
11980:978-0-13-285803-8
11973:. Prentice-Hall.
11929:, pp. 20–21.
11826:978-3-319-17770-0
11709:978-0-817-63250-2
11484:{\displaystyle X}
11462:Andrey Kolmogorov
11411:{\displaystyle A}
11349:{\displaystyle p}
11257:Suppose now that
11184:{\displaystyle p}
11073:
11067:
11061:
11055:
10978:
10799:{\displaystyle A}
10785:absorbing subsets
10782:absolutely convex
10773:{\displaystyle X}
10685:
10512:
10435:
10207:For instance, if
10095:{\displaystyle D}
10075:{\displaystyle C}
10047:{\displaystyle X}
9930:
9910:
9646:{\displaystyle p}
9592:{\displaystyle p}
9531:Illustrations of
9423:
9410:
9291:{\displaystyle u}
9267:{\displaystyle u}
9192:
8947:division algebras
8876:{\displaystyle z}
8856:{\displaystyle w}
8813:{\displaystyle N}
8464:
8369:{\displaystyle E}
8322:{\displaystyle k}
8269:{\displaystyle k}
8245:{\displaystyle E}
8088:{\displaystyle A}
8068:{\displaystyle A}
7993:{\displaystyle A}
7928:
7646:
7546:, and are called
7431:
7357:
7353:
7285:
7101:{\displaystyle p}
7081:{\displaystyle p}
7038:{\displaystyle x}
7002:signal processing
6775:
6708:with edge length
6409:
6359:
6322:
6267:
6139:
6063:
5981:
5882:
5848:{\displaystyle p}
5830:harmonic analysis
5773:{\displaystyle p}
5741:
5539:
5359:
5334:
5333:
5327:
5166:
4999:{\displaystyle p}
4945:
4902:{\displaystyle p}
4862:{\displaystyle p}
4614:-norm) of vector
4580:{\displaystyle p}
4258:
4060:
4054:
3992:
3983:
3945:
3918:
3794:{\displaystyle n}
3585:
3584:
3525:
3524:
3501:
3500:
3368:
3183:complex conjugate
3168:
3140:
3134:
3069:
2965:(also called the
2934:{\displaystyle n}
2816:distance function
2695:
2657:orthonormal basis
2554:
2366:{\displaystyle n}
2121:of vector spaces
2110:{\displaystyle X}
2090:{\displaystyle p}
1954:
1940:{\displaystyle 0}
1848:
1828:{\displaystyle X}
1701:{\displaystyle p}
1523:{\displaystyle q}
1503:{\displaystyle p}
1443:
1419:
1350:{\displaystyle q}
1337:is equivalent to
1330:{\displaystyle p}
1188:{\displaystyle C}
1168:{\displaystyle c}
1144:{\displaystyle q}
1124:{\displaystyle p}
1081:{\displaystyle q}
1061:{\displaystyle p}
949:{\displaystyle p}
894:{\displaystyle p}
836:{\displaystyle X}
352:{\displaystyle s}
261:{\displaystyle X}
212:{\displaystyle F}
189:{\displaystyle X}
16:(Redirected from
15180:
15140:
15139:
15114:Uniformly smooth
14783:
14775:
14742:Balanced/Circled
14732:Absorbing/Radial
14462:
14455:
14448:
14439:
14438:
14425:
14424:
14343:Jones polynomial
14261:Operator algebra
14005:
14004:
13978:
13971:
13964:
13955:
13954:
13907:
13905:
13904:
13899:
13894:
13893:
13861:Triebel–Lizorkin
13831:
13829:
13828:
13823:
13821:
13817:
13816:
13811:
13780:
13778:
13777:
13772:
13770:
13769:
13741:
13739:
13738:
13733:
13731:
13730:
13704:
13702:
13701:
13696:
13685:
13684:
13614:
13612:
13611:
13606:
13601:
13592:
13587:
13548:
13546:
13545:
13540:
13401:
13379:
13361:Balanced/Circled
13159:Robinson-Ursescu
13077:Eberlein–Šmulian
12997:Spectral theorem
12793:Linear operators
12590:Uniformly smooth
12487:
12480:
12473:
12464:
12463:
12459:
12434:Wilansky, Albert
12429:
12404:Trèves, François
12399:
12365:
12338:
12307:
12272:
12229:
12228:
12217:
12211:
12205:
12199:
12198:
12177:
12168:
12162:
12156:
12155:
12137:
12131:
12130:
12089:
12083:
12081:
12079:
12078:
12073:
12068:
12067:
12062:
12049:
12047:
12046:
12041:
12036:
12035:
12030:
12016:
12010:
12009:
12007:
12006:
11991:
11985:
11984:
11966:
11960:
11959:
11957:
11956:
11941:
11930:
11924:
11918:
11917:
11915:
11913:
11904:
11895:
11884:
11878:
11872:
11871:
11863:
11857:
11851:
11842:
11841:
11830:
11808:
11802:
11801:
11769:
11763:
11762:
11760:
11759:
11744:
11738:
11737:
11735:
11734:
11720:
11714:
11713:
11692:
11668:
11659:
11650:
11599:
11585:
11562:
11560:
11559:
11554:
11543:
11542:
11517:
11515:
11514:
11509:
11507:
11506:
11490:
11488:
11487:
11482:
11454:
11452:
11451:
11446:
11444:
11443:
11417:
11415:
11414:
11409:
11393:
11391:
11390:
11385:
11355:
11353:
11352:
11347:
11331:
11329:
11328:
11323:
11305:
11303:
11302:
11297:
11282:
11280:
11279:
11274:
11234:
11232:
11231:
11226:
11218:
11193:separates points
11190:
11188:
11187:
11182:
11170:
11168:
11167:
11162:
11133:
11131:
11130:
11125:
11120:
11116:
11100:
11099:
11071:
11065:
11059:
11053:
11052:
11048:
11032:
11031:
10994:
10992:
10991:
10986:
10983:
10970:
10968:
10967:
10962:
10930:
10901:
10900:
10884:
10882:
10881:
10876:
10855:
10853:
10852:
10847:
10845:
10844:
10828:
10826:
10825:
10820:
10805:
10803:
10802:
10797:
10779:
10777:
10776:
10771:
10737:
10735:
10734:
10729:
10724:
10723:
10702:
10701:
10686:
10681:
10676:
10675:
10657:
10656:
10638:
10637:
10615:
10613:
10612:
10607:
10602:
10601:
10580:
10579:
10561:
10560:
10539:
10537:
10536:
10531:
10529:
10528:
10513:
10508:
10503:
10502:
10484:
10483:
10462:
10460:
10459:
10454:
10452:
10451:
10436:
10431:
10426:
10425:
10407:
10406:
10382:
10380:
10379:
10374:
10369:
10368:
10353:
10352:
10345:
10331:
10312:
10311:
10293:
10292:
10270:
10268:
10267:
10262:
10257:
10256:
10251:
10238:
10236:
10235:
10230:
10206:
10204:
10203:
10198:
10193:
10192:
10171:
10170:
10152:
10151:
10127:
10125:
10124:
10119:
10101:
10099:
10098:
10093:
10081:
10079:
10078:
10073:
10060:
10059:
10053:
10051:
10050:
10045:
10033:
10031:
10030:
10025:
10023:
10022:
10000:
9998:
9997:
9992:
9990:
9989:
9965:
9963:
9962:
9957:
9931:
9928:
9911:
9908:
9848:
9846:
9845:
9840:
9806:
9804:
9803:
9798:
9777:
9775:
9774:
9769:
9761:
9757:
9750:
9749:
9728:
9726:
9725:
9720:
9701:
9699:
9698:
9693:
9688:
9687:
9652:
9650:
9649:
9644:
9632:
9630:
9629:
9624:
9598:
9596:
9595:
9590:
9539:Every norm is a
9524:
9522:
9521:
9516:
9511:
9510:
9495:
9494:
9476:
9472:
9440:
9438:
9437:
9432:
9424:
9416:
9411:
9403:
9400:
9399:
9384:
9383:
9365:
9345:
9328:
9326:
9325:
9320:
9318:
9317:
9297:
9295:
9294:
9289:
9273:
9271:
9270:
9265:
9253:
9251:
9250:
9245:
9221:
9219:
9218:
9213:
9193:
9190:
9188:
9156:
9119:
9117:
9116:
9111:
9096:
9094:
9093:
9088:
9086:
9042:
9040:
9039:
9034:
9032:
9020:
9018:
9017:
9012:
9007:
8996:
8994:
8993:
8988:
8983:
8972:
8970:
8969:
8964:
8959:
8944:
8942:
8941:
8936:
8882:
8880:
8879:
8874:
8862:
8860:
8859:
8854:
8843:of two elements
8842:
8840:
8839:
8834:
8819:
8817:
8816:
8811:
8791:
8789:
8788:
8783:
8781:
8780:
8742:
8740:
8739:
8734:
8729:
8728:
8723:
8710:
8708:
8707:
8702:
8684:
8682:
8681:
8676:
8665:
8664:
8659:
8628:
8626:
8625:
8620:
8591:
8589:
8588:
8585:{\displaystyle }
8583:
8558:
8556:
8555:
8552:{\displaystyle }
8550:
8525:
8523:
8522:
8517:
8512:
8511:
8510:
8509:
8499:
8495:
8494:
8484:
8483:
8472:
8450:
8448:
8447:
8442:
8420:
8418:
8417:
8412:
8407:
8406:
8401:
8397:
8396:
8375:
8373:
8372:
8367:
8352:If the distinct
8351:
8349:
8348:
8343:
8328:
8326:
8325:
8320:
8308:
8306:
8305:
8300:
8295:
8294:
8275:
8273:
8272:
8267:
8254:finite extension
8251:
8249:
8248:
8243:
8210:
8208:
8207:
8202:
8200:
8199:
8194:
8174:
8172:
8171:
8166:
8164:
8163:
8158:
8145:
8143:
8142:
8137:
8135:
8134:
8129:
8094:
8092:
8091:
8086:
8074:
8072:
8071:
8066:
8054:
8052:
8051:
8046:
8022:
8020:
8019:
8014:
7999:
7997:
7996:
7991:
7971:
7969:
7968:
7963:
7958:
7957:
7952:
7939:
7937:
7936:
7931:
7929:
7927:
7926:
7917:
7913:
7912:
7893:
7889:
7888:
7866:
7865:
7860:
7856:
7855:
7838:
7833:
7829:
7828:
7793:
7791:
7790:
7785:
7783:
7782:
7777:
7757:
7755:
7754:
7749:
7727:
7725:
7724:
7719:
7717:
7716:
7696:
7694:
7693:
7688:
7686:
7685:
7660:
7658:
7657:
7652:
7647:
7630:
7603:
7601:
7600:
7595:
7589:
7588:
7572:
7570:
7569:
7564:
7562:
7561:
7541:
7539:
7538:
7533:
7508:
7506:
7505:
7500:
7498:
7497:
7492:
7471:
7469:
7468:
7463:
7461:
7460:
7456:
7447:
7446:
7436:
7429:
7428:
7427:
7422:
7404:
7399:
7398:
7389:
7388:
7379:
7378:
7355:
7354:
7351:
7349:
7348:
7344:
7335:
7334:
7327:
7326:
7321:
7317:
7316:
7301:
7300:
7284:
7283:
7274:
7273:
7249:
7247:
7246:
7241:
7217:
7215:
7214:
7209:
7207:
7206:
7190:
7188:
7187:
7182:
7180:
7179:
7146:
7144:
7143:
7138:
7136:
7135:
7107:
7105:
7104:
7099:
7087:
7085:
7084:
7079:
7067:
7065:
7064:
7059:
7044:
7042:
7041:
7036:
7012:referred to the
6986:Hamming distance
6967:Hamming distance
6952:
6950:
6949:
6944:
6902:
6900:
6899:
6894:
6879:
6877:
6876:
6871:
6850:Here we mean by
6849:
6847:
6846:
6841:
6836:
6832:
6831:
6813:
6808:
6807:
6798:
6797:
6783:
6768:
6767:
6733:
6731:
6730:
6725:
6703:
6701:
6700:
6695:
6678:
6676:
6675:
6670:
6665:
6661:
6660:
6656:
6655:
6633:
6629:
6628:
6604:
6603:
6594:
6579:
6577:
6576:
6571:
6563:
6562:
6544:
6543:
6531:
6530:
6515:
6503:
6501:
6500:
6495:
6493:
6472:
6470:
6469:
6464:
6456:
6455:
6423:
6421:
6420:
6415:
6410:
6408:
6407:
6406:
6397:
6388:
6383:
6378:
6377:
6368:
6360:
6358:
6357:
6345:
6336:
6334:
6333:
6328:
6323:
6321:
6320:
6319:
6310:
6301:
6300:
6291:
6286:
6285:
6276:
6268:
6266:
6265:
6264:
6248:
6239:
6237:
6236:
6231:
6207:
6205:
6204:
6199:
6197:
6189:
6176:Hadamard product
6173:
6171:
6170:
6165:
6153:
6151:
6150:
6145:
6140:
6138:
6136:
6125:
6116:
6107:
6106:
6105:
6094:
6088:
6083:
6075:
6069:
6064:
6062:
6061:
6052:
6051:
6050:
6041:
6029:
6020:
6018:
6017:
6012:
5995:
5993:
5992:
5987:
5982:
5980:
5978:
5967:
5958:
5949:
5948:
5947:
5936:
5932:
5931:
5917:
5916:
5906:
5901:
5900:
5891:
5883:
5881:
5880:
5879:
5863:
5854:
5852:
5851:
5846:
5815:
5813:
5812:
5807:
5796:
5795:
5779:
5777:
5776:
5771:
5759:
5757:
5756:
5751:
5746:
5739:
5738:
5737:
5732:
5699:
5694:
5693:
5677:
5675:
5674:
5669:
5667:
5666:
5650:
5648:
5647:
5642:
5611:
5609:
5608:
5603:
5574:
5572:
5571:
5566:
5558:
5540:
5535:
5521:
5519:
5518:
5506:
5505:
5504:
5503:
5466:
5464:
5463:
5458:
5422:
5420:
5419:
5414:
5397:
5396:
5380:
5378:
5377:
5372:
5370:
5369:
5360:
5355:
5354:
5345:
5342:
5331:
5325:
5324:
5323:
5322:
5321:
5307:
5306:
5301:
5297:
5296:
5279:
5278:
5273:
5269:
5268:
5243:
5242:
5236:
5234:
5233:
5228:
5223:
5222:
5202:
5200:
5199:
5194:
5189:
5178:for all vectors
5177:
5175:
5174:
5169:
5167:
5162:
5154:
5146:
5141:
5140:
5131:
5116:
5114:
5113:
5108:
5076:
5074:
5073:
5068:
5066:
5065:
5041:
5039:
5038:
5033:
5008:generalized mean
5005:
5003:
5002:
4997:
4985:
4983:
4982:
4977:
4972:
4968:
4967:
4953:
4941:
4940:
4931:
4908:
4906:
4905:
4900:
4888:
4886:
4885:
4880:
4868:
4866:
4865:
4860:
4844:
4842:
4841:
4836:
4814:
4812:
4811:
4806:
4785:
4783:
4782:
4777:
4772:
4771:
4767:
4758:
4754:
4753:
4752:
4747:
4743:
4742:
4727:
4722:
4698:
4697:
4688:
4673:
4671:
4670:
4665:
4660:
4659:
4641:
4640:
4625:
4613:
4611:
4610:
4605:
4603:
4602:
4586:
4584:
4583:
4578:
4566:
4564:
4563:
4558:
4523:
4521:
4520:
4515:
4513:
4512:
4502:
4497:
4467:
4465:
4464:
4459:
4457:
4456:
4433:
4431:
4430:
4425:
4423:
4422:
4399:
4397:
4396:
4391:
4364:
4362:
4361:
4356:
4351:
4347:
4346:
4332:
4327:
4309:
4308:
4299:
4282:Taxicab geometry
4272:
4270:
4269:
4264:
4259:
4257:
4249:
4244:
4236:
4206:
4204:
4203:
4198:
4196:
4195:
4190:
4177:
4175:
4174:
4169:
4167:
4166:
4165:
4159:
4158:
4151:
4150:
4136:
4135:
4125:
4124:
4096:
4094:
4093:
4088:
4086:
4074:
4072:
4071:
4066:
4061:
4059:
4052:
4051:
4050:
4045:
4039:
4031:
4006:
4004:
4003:
3998:
3993:
3991:
3990:
3985:
3984:
3976:
3972:
3971:
3953:
3952:
3947:
3946:
3938:
3934:
3933:
3924:
3919:
3917:
3916:
3911:
3907:
3906:
3883:
3882:
3877:
3873:
3872:
3858:
3850:
3835:
3833:
3832:
3827:
3822:
3821:
3816:
3800:
3798:
3797:
3792:
3773:
3771:
3770:
3765:
3760:
3759:
3754:
3737:
3735:
3734:
3729:
3724:
3723:
3718:
3705:
3703:
3702:
3697:
3695:
3683:
3681:
3680:
3675:
3670:
3658:
3656:
3655:
3650:
3648:
3636:
3624:
3596:
3594:
3593:
3588:
3586:
3582:
3581:
3580:
3568:
3567:
3555:
3554:
3542:
3541:
3531:
3526:
3522:
3518:
3517:
3507:
3502:
3498:
3497:
3496:
3483:
3458:
3456:
3455:
3450:
3448:
3429:
3427:
3426:
3421:
3419:
3407:
3405:
3404:
3399:
3397:
3385:
3383:
3382:
3377:
3369:
3366:
3339:
3337:
3336:
3331:
3326:
3311:
3309:
3308:
3303:
3298:
3283:
3281:
3280:
3275:
3270:
3258:
3256:
3255:
3250:
3245:
3208:
3206:
3205:
3200:
3180:
3178:
3177:
3172:
3170:
3169:
3161:
3151:
3149:
3148:
3143:
3141:
3136:
3135:
3127:
3124:
3115:
3113:
3112:
3107:
3080:
3078:
3077:
3072:
3070:
3068:
3067:
3055:
3054:
3045:
3036:
3034:
3033:
3028:
3007:
3005:
3004:
2999:
2994:
2993:
2988:
2969:) of it, if the
2940:
2938:
2937:
2932:
2919:
2917:
2916:
2911:
2909:
2908:
2897:
2879:
2877:
2876:
2871:
2869:
2868:
2849:
2847:
2846:
2841:
2839:
2838:
2820:Euclidean length
2811:
2809:
2808:
2803:
2801:
2800:
2773:
2771:
2770:
2765:
2763:
2762:
2743:
2741:
2740:
2735:
2733:
2732:
2709:
2707:
2706:
2701:
2696:
2694:
2686:
2681:
2673:
2635:
2633:
2632:
2627:
2622:
2621:
2616:
2568:
2566:
2565:
2560:
2555:
2552:
2547:
2528:
2523:
2514:
2509:
2508:
2499:
2484:
2482:
2481:
2476:
2474:
2470:
2469:
2468:
2450:
2449:
2437:
2436:
2419:
2407:
2405:
2404:
2399:
2394:
2393:
2388:
2372:
2370:
2369:
2364:
2334:
2332:
2331:
2326:
2311:
2309:
2308:
2303:
2301:
2283:
2281:
2280:
2275:
2246:
2238:
2226:
2224:
2223:
2218:
2213:
2201:
2199:
2198:
2193:
2191:
2179:
2177:
2176:
2171:
2169:
2157:
2155:
2154:
2149:
2138:
2116:
2114:
2113:
2108:
2096:
2094:
2093:
2088:
2061:
2059:
2058:
2053:
2051:
2043:
2020:
2018:
2017:
2012:
1997:
1995:
1994:
1989:
1987:
1983:
1982:
1968:
1946:
1944:
1943:
1938:
1926:
1924:
1923:
1918:
1916:
1915:
1899:
1897:
1896:
1891:
1883:
1882:
1873:
1872:
1862:
1834:
1832:
1831:
1826:
1810:
1808:
1807:
1802:
1800:
1799:
1788:
1784:
1783:
1766:
1765:
1741:
1739:
1738:
1733:
1731:
1723:
1712:), the notation
1707:
1705:
1704:
1699:
1687:
1685:
1684:
1679:
1643:
1641:
1640:
1635:
1617:
1615:
1614:
1609:
1594:
1592:
1591:
1586:
1584:
1552:
1550:
1549:
1544:
1529:
1527:
1526:
1521:
1509:
1507:
1506:
1501:
1489:
1487:
1486:
1481:
1458:
1456:
1455:
1450:
1445:
1436:
1421:
1412:
1402:
1400:
1399:
1394:
1356:
1354:
1353:
1348:
1336:
1334:
1333:
1328:
1316:
1314:
1313:
1308:
1249:
1247:
1246:
1241:
1220:
1218:
1217:
1212:
1194:
1192:
1191:
1186:
1174:
1172:
1171:
1166:
1150:
1148:
1147:
1142:
1130:
1128:
1127:
1122:
1110:
1108:
1107:
1102:
1087:
1085:
1084:
1079:
1067:
1065:
1064:
1059:
1044:Equivalent norms
1028:
1026:
1025:
1020:
999:
997:
996:
991:
955:
953:
952:
947:
935:
933:
932:
927:
900:
898:
897:
892:
876:
874:
873:
868:
866:
842:
840:
839:
834:
813:
811:
810:
805:
787:
785:
784:
779:
753:
751:
750:
745:
724:
722:
721:
716:
684:
682:
681:
676:
658:
656:
655:
650:
623:
621:
620:
615:
594:
593:
592:Point-separating
583:
581:
580:
575:
561:and all scalars
560:
558:
557:
552:
534:
532:
531:
526:
512:
504:
469:
467:
466:
461:
434:
432:
431:
426:
358:
356:
355:
350:
334:
332:
331:
326:
324:
316:
304:
302:
301:
296:
294:
267:
265:
264:
259:
243:
241:
240:
235:
230:
218:
216:
215:
210:
195:
193:
192:
187:
160:
158:
157:
152:
21:
15188:
15187:
15183:
15182:
15181:
15179:
15178:
15177:
15153:
15152:
15151:
15146:
15128:
14890:B-complete/Ptak
14873:
14817:
14781:
14773:
14752:Bounding points
14715:
14657:Densely defined
14603:
14592:Bounded inverse
14538:
14472:
14466:
14436:
14431:
14413:
14377:Advanced topics
14372:
14296:
14275:
14234:
14200:Hilbert–Schmidt
14173:
14164:Gelfand–Naimark
14111:
14061:
13996:
13982:
13952:
13947:
13911:
13889:
13885:
13871:
13868:
13867:
13866:Wiener amalgam
13836:Segal–Bargmann
13812:
13807:
13806:
13802:
13797:
13794:
13793:
13765:
13761:
13759:
13756:
13755:
13726:
13722:
13720:
13717:
13716:
13674:
13670:
13668:
13665:
13664:
13619:Birnbaum–Orlicz
13597:
13588:
13577:
13571:
13568:
13567:
13522:
13519:
13518:
13496:
13452:Bounding points
13425:
13399:
13377:
13334:
13185:Banach manifold
13168:
13092:Gelfand–Naimark
13013:
12987:Spectral theory
12955:Banach algebras
12947:Operator theory
12941:
12902:Pseudo-monotone
12885:Hilbert–Schmidt
12865:Densely defined
12787:
12700:
12614:
12497:
12491:
12448:
12418:
12388:
12354:
12327:
12296:
12286:Springer-Verlag
12261:
12238:
12233:
12232:
12219:
12218:
12214:
12206:
12202:
12195:
12178:
12171:
12163:
12159:
12152:
12138:
12134:
12112:
12090:
12086:
12063:
12058:
12057:
12055:
12052:
12051:
12031:
12026:
12025:
12023:
12020:
12019:
12017:
12013:
12004:
12002:
11992:
11988:
11981:
11967:
11963:
11954:
11952:
11942:
11933:
11925:
11921:
11911:
11909:
11902:
11898:Conrad, Keith.
11896:
11887:
11879:
11875:
11864:
11860:
11852:
11845:
11827:
11809:
11805:
11770:
11766:
11757:
11755:
11752:www.spektrum.de
11746:
11745:
11741:
11732:
11730:
11722:
11721:
11717:
11710:
11693:
11686:
11681:
11666:
11657:
11648:
11597:
11583:
11574:Asymmetric norm
11570:
11538:
11534:
11523:
11520:
11519:
11502:
11498:
11496:
11493:
11492:
11476:
11473:
11472:
11439:
11435:
11427:
11424:
11423:
11403:
11400:
11399:
11361:
11358:
11357:
11356:is a norm, and
11341:
11338:
11337:
11311:
11308:
11307:
11288:
11285:
11284:
11262:
11259:
11258:
11214:
11200:
11197:
11196:
11176:
11173:
11172:
11150:
11147:
11146:
11095:
11091:
11078:
11074:
11027:
11023:
11010:
11006:
11004:
11001:
11000:
10982:
10976:
10973:
10972:
10926:
10896:
10892:
10890:
10887:
10886:
10867:
10864:
10863:
10840:
10836:
10834:
10831:
10830:
10811:
10808:
10807:
10791:
10788:
10787:
10765:
10762:
10761:
10758:
10752:
10719:
10715:
10697:
10693:
10680:
10671:
10667:
10652:
10648:
10633:
10629:
10621:
10618:
10617:
10597:
10593:
10575:
10571:
10556:
10552:
10544:
10541:
10540:
10524:
10520:
10507:
10498:
10494:
10479:
10475:
10467:
10464:
10463:
10447:
10443:
10430:
10421:
10417:
10402:
10398:
10390:
10387:
10386:
10385:In particular,
10364:
10360:
10341:
10327:
10320:
10316:
10307:
10303:
10288:
10284:
10276:
10273:
10272:
10252:
10247:
10246:
10244:
10241:
10240:
10212:
10209:
10208:
10188:
10184:
10166:
10162:
10147:
10143:
10132:
10129:
10128:
10107:
10104:
10103:
10087:
10084:
10083:
10067:
10064:
10063:
10057:
10056:
10039:
10036:
10035:
10018:
10014:
10006:
10003:
10002:
9985:
9981:
9973:
9970:
9969:
9929: and
9927:
9907:
9854:
9851:
9850:
9816:
9813:
9812:
9783:
9780:
9779:
9745:
9741:
9740:
9736:
9734:
9731:
9730:
9711:
9708:
9707:
9683:
9679:
9674:
9671:
9670:
9638:
9635:
9634:
9612:
9609:
9608:
9599:-norm, it is a
9584:
9581:
9580:
9567:The concept of
9565:
9557:convex function
9506:
9502:
9490:
9486:
9456:
9452:
9450:
9447:
9446:
9415:
9402:
9395:
9391:
9379:
9375:
9361:
9341:
9339:
9336:
9335:
9313:
9309:
9307:
9304:
9303:
9283:
9280:
9279:
9259:
9256:
9255:
9227:
9224:
9223:
9189:
9184:
9152:
9129:
9126:
9125:
9102:
9099:
9098:
9082:
9068:
9065:
9064:
9061:
9051:the norm is an
9028:
9026:
9023:
9022:
9003:
9001:
8998:
8997:
8979:
8977:
8974:
8973:
8955:
8953:
8950:
8949:
8945:In the case of
8888:
8885:
8884:
8868:
8865:
8864:
8848:
8845:
8844:
8825:
8822:
8821:
8805:
8802:
8801:
8776:
8772:
8752:
8749:
8748:
8724:
8722:
8721:
8719:
8716:
8715:
8693:
8690:
8689:
8685:consists of an
8660:
8658:
8657:
8646:
8643:
8642:
8605:
8602:
8601:
8598:
8565:
8562:
8561:
8532:
8529:
8528:
8505:
8501:
8500:
8479:
8475:
8474:
8468:
8463:
8459:
8458:
8456:
8453:
8452:
8430:
8427:
8426:
8402:
8392:
8388:
8384:
8383:
8381:
8378:
8377:
8361:
8358:
8357:
8334:
8331:
8330:
8314:
8311:
8310:
8290:
8286:
8284:
8281:
8280:
8261:
8258:
8257:
8237:
8234:
8233:
8230:
8224:
8195:
8190:
8189:
8187:
8184:
8183:
8159:
8154:
8153:
8151:
8148:
8147:
8130:
8125:
8124:
8122:
8119:
8118:
8080:
8077:
8076:
8060:
8057:
8056:
8028:
8025:
8024:
8005:
8002:
8001:
7985:
7982:
7981:
7953:
7948:
7947:
7945:
7942:
7941:
7922:
7918:
7908:
7904:
7900:
7884:
7880:
7876:
7861:
7851:
7847:
7843:
7842:
7837:
7824:
7820:
7816:
7799:
7796:
7795:
7778:
7773:
7772:
7770:
7767:
7766:
7765:Other norms on
7763:
7761:Composite norms
7733:
7730:
7729:
7712:
7708:
7706:
7703:
7702:
7681:
7677:
7675:
7672:
7671:
7629:
7615:
7612:
7611:
7584:
7580:
7578:
7575:
7574:
7557:
7553:
7551:
7548:
7547:
7518:
7515:
7514:
7493:
7488:
7487:
7479:
7476:
7475:
7452:
7448:
7442:
7441:
7440:
7432:
7423:
7418:
7417:
7400:
7394:
7390:
7384:
7383:
7368:
7364:
7352: and
7350:
7340:
7336:
7330:
7329:
7328:
7322:
7312:
7308:
7304:
7303:
7296:
7289:
7279:
7278:
7269:
7265:
7257:
7254:
7253:
7225:
7222:
7221:
7202:
7198:
7196:
7193:
7192:
7175:
7171:
7169:
7166:
7165:
7161:
7131:
7127:
7125:
7122:
7121:
7093:
7090:
7089:
7073:
7070:
7069:
7050:
7047:
7046:
7030:
7027:
7026:
6981:discrete metric
6977:metric geometry
6973:
6971:discrete metric
6963:
6908:
6905:
6904:
6885:
6882:
6881:
6859:
6856:
6855:
6827:
6823:
6809:
6803:
6799:
6790:
6786:
6785:
6779:
6763:
6759:
6754:
6751:
6750:
6739:
6713:
6710:
6709:
6686:
6683:
6682:
6651:
6647:
6643:
6624:
6620:
6616:
6615:
6611:
6599:
6595:
6590:
6585:
6582:
6581:
6558:
6554:
6539:
6535:
6526:
6522:
6511:
6509:
6506:
6505:
6489:
6487:
6484:
6483:
6480:
6451:
6447:
6439:
6436:
6435:
6429:
6402:
6398:
6393:
6389:
6384:
6382:
6373:
6369:
6364:
6353:
6349:
6344:
6342:
6339:
6338:
6315:
6311:
6306:
6302:
6296:
6292:
6290:
6281:
6277:
6272:
6260:
6256:
6252:
6247:
6245:
6242:
6241:
6216:
6213:
6212:
6193:
6185:
6183:
6180:
6179:
6159:
6156:
6155:
6126:
6121:
6112:
6108:
6095:
6090:
6089:
6084:
6079:
6071:
6070:
6068:
6057:
6053:
6046:
6042:
6037:
6030:
6028:
6026:
6023:
6022:
6003:
6000:
5999:
5968:
5963:
5954:
5950:
5937:
5927:
5923:
5919:
5918:
5912:
5908:
5907:
5905:
5896:
5892:
5887:
5875:
5871:
5867:
5862:
5860:
5857:
5856:
5840:
5837:
5836:
5791:
5787:
5785:
5782:
5781:
5765:
5762:
5761:
5742:
5733:
5728:
5727:
5695:
5689:
5685:
5683:
5680:
5679:
5662:
5658:
5656:
5653:
5652:
5621:
5618:
5617:
5582:
5579:
5578:
5554:
5522:
5520:
5514:
5510:
5499:
5495:
5494:
5490:
5476:
5473:
5472:
5431:
5428:
5427:
5392:
5388:
5386:
5383:
5382:
5365:
5361:
5350:
5346:
5344:
5338:
5317:
5313:
5312:
5308:
5302:
5292:
5288:
5284:
5283:
5274:
5264:
5260:
5256:
5255:
5250:
5247:
5246:
5240:
5239:
5218:
5214:
5212:
5209:
5208:
5185:
5183:
5180:
5179:
5158:
5150:
5145:
5136:
5132:
5127:
5122:
5119:
5118:
5085:
5082:
5081:
5061:
5057:
5047:
5044:
5043:
5018:
5015:
5014:
5010:or power mean.
4991:
4988:
4987:
4963:
4959:
4955:
4949:
4936:
4932:
4927:
4922:
4919:
4918:
4894:
4891:
4890:
4874:
4871:
4870:
4854:
4851:
4850:
4824:
4821:
4820:
4791:
4788:
4787:
4763:
4759:
4748:
4738:
4734:
4730:
4729:
4723:
4712:
4707:
4703:
4702:
4693:
4689:
4684:
4679:
4676:
4675:
4655:
4651:
4636:
4632:
4621:
4619:
4616:
4615:
4598:
4594:
4592:
4589:
4588:
4572:
4569:
4568:
4546:
4543:
4542:
4539:
4533:
4508:
4504:
4498:
4487:
4481:
4478:
4477:
4452:
4448:
4446:
4443:
4442:
4418:
4414:
4412:
4409:
4408:
4382:
4379:
4378:
4342:
4338:
4334:
4328:
4317:
4304:
4300:
4295:
4290:
4287:
4286:
4284:
4278:
4253:
4245:
4243:
4232:
4227:
4224:
4223:
4191:
4186:
4185:
4183:
4180:
4179:
4161:
4160:
4153:
4152:
4146:
4142:
4131:
4127:
4120:
4116:
4109:
4108:
4107:
4105:
4102:
4101:
4082:
4080:
4077:
4076:
4055:
4046:
4041:
4040:
4038:
4027:
4022:
4019:
4018:
3986:
3975:
3974:
3973:
3967:
3963:
3948:
3937:
3936:
3935:
3929:
3925:
3923:
3912:
3902:
3898:
3894:
3893:
3878:
3868:
3864:
3860:
3859:
3857:
3846:
3841:
3838:
3837:
3817:
3812:
3811:
3809:
3806:
3805:
3786:
3783:
3782:
3779:
3755:
3750:
3749:
3747:
3744:
3743:
3719:
3714:
3713:
3711:
3708:
3707:
3691:
3689:
3686:
3685:
3666:
3664:
3661:
3660:
3644:
3632:
3620:
3602:
3599:
3598:
3576:
3572:
3563:
3559:
3550:
3546:
3537:
3533:
3530:
3513:
3509:
3506:
3492:
3488:
3482:
3468:
3465:
3464:
3444:
3442:
3439:
3438:
3415:
3413:
3410:
3409:
3393:
3391:
3388:
3387:
3367: and
3365:
3345:
3342:
3341:
3322:
3320:
3317:
3316:
3312:and lastly the
3294:
3292:
3289:
3288:
3266:
3264:
3261:
3260:
3241:
3239:
3236:
3235:
3224:
3214:
3190:
3187:
3186:
3160:
3159:
3157:
3154:
3153:
3126:
3125:
3123:
3121:
3118:
3117:
3086:
3083:
3082:
3063:
3059:
3050:
3046:
3044:
3042:
3039:
3038:
3013:
3010:
3009:
2989:
2984:
2983:
2981:
2978:
2977:
2975:Euclidean plane
2955:
2949:
2926:
2923:
2922:
2898:
2893:
2892:
2890:
2887:
2886:
2864:
2860:
2858:
2855:
2854:
2834:
2830:
2828:
2825:
2824:
2814:. It defines a
2796:
2792:
2790:
2787:
2786:
2758:
2754:
2752:
2749:
2748:
2728:
2724:
2722:
2719:
2718:
2690:
2682:
2680:
2669:
2664:
2661:
2660:
2617:
2612:
2611:
2609:
2606:
2605:
2548:
2543:
2524:
2519:
2513:
2504:
2500:
2495:
2490:
2487:
2486:
2464:
2460:
2445:
2441:
2432:
2428:
2427:
2423:
2415:
2413:
2410:
2409:
2389:
2384:
2383:
2381:
2378:
2377:
2375:Euclidean space
2358:
2355:
2354:
2351:
2341:
2317:
2314:
2313:
2297:
2289:
2286:
2285:
2242:
2234:
2232:
2229:
2228:
2209:
2207:
2204:
2203:
2187:
2185:
2182:
2181:
2165:
2163:
2160:
2159:
2134:
2126:
2123:
2122:
2102:
2099:
2098:
2082:
2079:
2078:
2068:complex numbers
2047:
2039:
2037:
2034:
2033:
2027:
2003:
2000:
1999:
1978:
1974:
1970:
1958:
1952:
1949:
1948:
1932:
1929:
1928:
1911:
1907:
1905:
1902:
1901:
1878:
1874:
1868:
1864:
1852:
1840:
1837:
1836:
1820:
1817:
1816:
1789:
1779:
1775:
1771:
1770:
1761:
1757:
1755:
1752:
1751:
1748:
1727:
1719:
1717:
1714:
1713:
1710:explained below
1693:
1690:
1689:
1649:
1646:
1645:
1623:
1620:
1619:
1600:
1597:
1596:
1580:
1566:
1563:
1562:
1559:
1535:
1532:
1531:
1515:
1512:
1511:
1495:
1492:
1491:
1472:
1469:
1468:
1434:
1410:
1408:
1405:
1404:
1370:
1367:
1366:
1342:
1339:
1338:
1322:
1319:
1318:
1254:
1251:
1250:
1226:
1223:
1222:
1200:
1197:
1196:
1180:
1177:
1176:
1160:
1157:
1156:
1136:
1133:
1132:
1116:
1113:
1112:
1093:
1090:
1089:
1073:
1070:
1069:
1053:
1050:
1049:
1046:
1005:
1002:
1001:
970:
967:
966:
941:
938:
937:
906:
903:
902:
886:
883:
882:
862:
848:
845:
844:
828:
825:
824:
793:
790:
789:
788:if and only if
758:
755:
754:
730:
727:
726:
692:
689:
688:
664:
661:
660:
629:
626:
625:
600:
597:
596:
591:
590:
566:
563:
562:
540:
537:
536:
508:
500:
480:
477:
476:
440:
437:
436:
375:
372:
371:
344:
341:
340:
320:
312:
310:
307:
306:
290:
276:
273:
272:
253:
250:
249:
226:
224:
221:
220:
204:
201:
200:
181:
178:
177:
171:
144:
141:
140:
87:Euclidean space
51:
48:prewellordering
28:
23:
22:
15:
12:
11:
5:
15186:
15176:
15175:
15170:
15168:Linear algebra
15165:
15148:
15147:
15145:
15144:
15133:
15130:
15129:
15127:
15126:
15121:
15116:
15111:
15109:Ultrabarrelled
15101:
15095:
15090:
15084:
15079:
15074:
15069:
15064:
15059:
15050:
15044:
15039:
15037:Quasi-complete
15034:
15032:Quasibarrelled
15029:
15024:
15019:
15014:
15009:
15004:
14999:
14994:
14989:
14984:
14979:
14974:
14973:
14972:
14962:
14957:
14952:
14947:
14942:
14937:
14932:
14927:
14922:
14912:
14907:
14897:
14892:
14887:
14881:
14879:
14875:
14874:
14872:
14871:
14861:
14856:
14851:
14846:
14841:
14831:
14825:
14823:
14822:Set operations
14819:
14818:
14816:
14815:
14810:
14805:
14800:
14795:
14790:
14785:
14777:
14769:
14764:
14759:
14754:
14749:
14744:
14739:
14734:
14729:
14723:
14721:
14717:
14716:
14714:
14713:
14708:
14703:
14698:
14693:
14692:
14691:
14686:
14681:
14671:
14666:
14665:
14664:
14659:
14654:
14649:
14644:
14639:
14634:
14624:
14623:
14622:
14611:
14609:
14605:
14604:
14602:
14601:
14596:
14595:
14594:
14584:
14578:
14569:
14564:
14559:
14557:Banach–Alaoglu
14554:
14552:Anderson–Kadec
14548:
14546:
14540:
14539:
14537:
14536:
14531:
14526:
14521:
14516:
14511:
14506:
14501:
14496:
14491:
14486:
14480:
14478:
14477:Basic concepts
14474:
14473:
14465:
14464:
14457:
14450:
14442:
14433:
14432:
14430:
14429:
14418:
14415:
14414:
14412:
14411:
14406:
14401:
14396:
14394:Choquet theory
14391:
14386:
14380:
14378:
14374:
14373:
14371:
14370:
14360:
14355:
14350:
14345:
14340:
14335:
14330:
14325:
14320:
14315:
14310:
14304:
14302:
14298:
14297:
14295:
14294:
14289:
14283:
14281:
14277:
14276:
14274:
14273:
14268:
14263:
14258:
14253:
14248:
14246:Banach algebra
14242:
14240:
14236:
14235:
14233:
14232:
14227:
14222:
14217:
14212:
14207:
14202:
14197:
14192:
14187:
14181:
14179:
14175:
14174:
14172:
14171:
14169:Banach–Alaoglu
14166:
14161:
14156:
14151:
14146:
14141:
14136:
14131:
14125:
14123:
14117:
14116:
14113:
14112:
14110:
14109:
14104:
14099:
14097:Locally convex
14094:
14080:
14075:
14069:
14067:
14063:
14062:
14060:
14059:
14054:
14049:
14044:
14039:
14034:
14029:
14024:
14019:
14014:
14008:
14002:
13998:
13997:
13981:
13980:
13973:
13966:
13958:
13949:
13948:
13946:
13945:
13940:
13935:
13930:
13925:
13919:
13917:
13913:
13912:
13910:
13909:
13897:
13892:
13888:
13884:
13881:
13878:
13875:
13863:
13858:
13857:
13856:
13846:
13844:Sequence space
13841:
13833:
13820:
13815:
13810:
13805:
13801:
13789:
13788:
13787:
13782:
13768:
13764:
13745:
13744:
13743:
13729:
13725:
13706:
13694:
13691:
13688:
13683:
13680:
13677:
13673:
13660:
13652:
13647:
13634:
13629:
13621:
13616:
13604:
13600:
13596:
13591:
13586:
13583:
13580:
13576:
13563:
13555:
13550:
13538:
13535:
13532:
13529:
13526:
13515:
13506:
13504:
13498:
13497:
13495:
13494:
13484:
13479:
13474:
13469:
13464:
13459:
13454:
13449:
13439:
13433:
13431:
13427:
13426:
13424:
13423:
13418:
13413:
13408:
13403:
13395:
13381:
13373:
13368:
13363:
13358:
13353:
13348:
13342:
13340:
13336:
13335:
13333:
13332:
13322:
13321:
13320:
13315:
13310:
13300:
13299:
13298:
13293:
13288:
13278:
13277:
13276:
13271:
13266:
13261:
13259:Gelfand–Pettis
13256:
13251:
13241:
13240:
13239:
13234:
13229:
13224:
13219:
13209:
13204:
13199:
13194:
13193:
13192:
13182:
13176:
13174:
13170:
13169:
13167:
13166:
13161:
13156:
13151:
13146:
13141:
13136:
13131:
13126:
13121:
13116:
13111:
13110:
13109:
13099:
13094:
13089:
13084:
13079:
13074:
13069:
13064:
13059:
13054:
13049:
13044:
13039:
13034:
13032:Banach–Alaoglu
13029:
13027:Anderson–Kadec
13023:
13021:
13015:
13014:
13012:
13011:
13006:
13001:
13000:
12999:
12994:
12984:
12983:
12982:
12977:
12967:
12965:Operator space
12962:
12957:
12951:
12949:
12943:
12942:
12940:
12939:
12934:
12929:
12924:
12919:
12914:
12909:
12904:
12899:
12898:
12897:
12887:
12882:
12881:
12880:
12875:
12867:
12862:
12852:
12851:
12850:
12840:
12835:
12825:
12824:
12823:
12818:
12813:
12803:
12797:
12795:
12789:
12788:
12786:
12785:
12780:
12775:
12774:
12773:
12768:
12758:
12757:
12756:
12751:
12741:
12736:
12731:
12730:
12729:
12719:
12714:
12708:
12706:
12702:
12701:
12699:
12698:
12693:
12688:
12687:
12686:
12676:
12671:
12666:
12665:
12664:
12653:Locally convex
12650:
12649:
12648:
12638:
12633:
12628:
12622:
12620:
12616:
12615:
12613:
12612:
12605:Tensor product
12598:
12592:
12587:
12581:
12576:
12570:
12565:
12560:
12550:
12549:
12548:
12543:
12533:
12528:
12526:Banach lattice
12523:
12522:
12521:
12511:
12505:
12503:
12499:
12498:
12490:
12489:
12482:
12475:
12467:
12461:
12460:
12446:
12430:
12416:
12400:
12386:
12366:
12353:978-1584888666
12352:
12339:
12325:
12308:
12294:
12273:
12259:
12237:
12234:
12231:
12230:
12212:
12200:
12193:
12169:
12157:
12150:
12132:
12110:
12084:
12071:
12066:
12061:
12039:
12034:
12029:
12011:
11986:
11979:
11961:
11931:
11919:
11885:
11873:
11858:
11856:, p. 200.
11843:
11825:
11803:
11764:
11739:
11715:
11708:
11683:
11682:
11680:
11677:
11676:
11675:
11669:
11660:
11651:
11642:
11636:
11630:
11624:
11618:
11612:
11606:
11600:
11591:
11586:
11577:
11569:
11566:
11565:
11564:
11552:
11549:
11546:
11541:
11537:
11533:
11530:
11527:
11505:
11501:
11480:
11469:
11457:
11456:
11455:is continuous.
11442:
11438:
11434:
11431:
11407:
11383:
11380:
11377:
11374:
11371:
11368:
11365:
11345:
11321:
11318:
11315:
11295:
11292:
11272:
11269:
11266:
11224:
11221:
11217:
11213:
11210:
11207:
11204:
11180:
11160:
11157:
11154:
11123:
11119:
11115:
11112:
11109:
11106:
11103:
11098:
11094:
11090:
11087:
11084:
11081:
11077:
11070:
11064:
11058:
11051:
11047:
11044:
11041:
11038:
11035:
11030:
11026:
11022:
11019:
11016:
11013:
11009:
10981:
10960:
10957:
10954:
10951:
10948:
10945:
10942:
10939:
10936:
10933:
10929:
10925:
10922:
10919:
10916:
10913:
10910:
10907:
10904:
10899:
10895:
10874:
10871:
10843:
10839:
10818:
10815:
10795:
10769:
10754:Main article:
10751:
10748:
10727:
10722:
10718:
10714:
10711:
10708:
10705:
10700:
10696:
10692:
10689:
10684:
10679:
10674:
10670:
10666:
10663:
10660:
10655:
10651:
10647:
10644:
10641:
10636:
10632:
10628:
10625:
10605:
10600:
10596:
10592:
10589:
10586:
10583:
10578:
10574:
10570:
10567:
10564:
10559:
10555:
10551:
10548:
10527:
10523:
10519:
10516:
10511:
10506:
10501:
10497:
10493:
10490:
10487:
10482:
10478:
10474:
10471:
10450:
10446:
10442:
10439:
10434:
10429:
10424:
10420:
10416:
10413:
10410:
10405:
10401:
10397:
10394:
10372:
10367:
10363:
10359:
10356:
10351:
10348:
10344:
10340:
10337:
10334:
10330:
10326:
10323:
10319:
10315:
10310:
10306:
10302:
10299:
10296:
10291:
10287:
10283:
10280:
10260:
10255:
10250:
10228:
10225:
10222:
10219:
10216:
10196:
10191:
10187:
10183:
10180:
10177:
10174:
10169:
10165:
10161:
10158:
10155:
10150:
10146:
10142:
10139:
10136:
10117:
10114:
10111:
10091:
10071:
10043:
10021:
10017:
10013:
10010:
9988:
9984:
9980:
9977:
9955:
9952:
9949:
9946:
9943:
9940:
9937:
9934:
9926:
9923:
9920:
9917:
9914:
9906:
9903:
9900:
9897:
9894:
9891:
9888:
9885:
9882:
9879:
9876:
9873:
9870:
9867:
9864:
9861:
9858:
9838:
9835:
9832:
9829:
9826:
9823:
9820:
9796:
9793:
9790:
9787:
9767:
9764:
9760:
9756:
9753:
9748:
9744:
9739:
9718:
9715:
9691:
9686:
9682:
9678:
9642:
9622:
9619:
9616:
9588:
9564:
9561:
9514:
9509:
9505:
9501:
9498:
9493:
9489:
9485:
9482:
9479:
9475:
9471:
9468:
9465:
9462:
9459:
9455:
9430:
9427:
9422:
9419:
9414:
9409:
9406:
9398:
9394:
9390:
9387:
9382:
9378:
9374:
9371:
9368:
9364:
9360:
9357:
9354:
9351:
9348:
9344:
9316:
9312:
9287:
9263:
9243:
9240:
9237:
9234:
9231:
9211:
9208:
9205:
9202:
9199:
9196:
9187:
9183:
9180:
9177:
9174:
9171:
9168:
9165:
9162:
9159:
9155:
9151:
9148:
9145:
9142:
9139:
9136:
9133:
9109:
9106:
9085:
9081:
9078:
9075:
9072:
9060:
9057:
9049:split algebras
9031:
9010:
9006:
8986:
8982:
8962:
8958:
8934:
8931:
8928:
8925:
8922:
8919:
8916:
8913:
8910:
8907:
8904:
8901:
8898:
8895:
8892:
8872:
8852:
8832:
8829:
8809:
8779:
8775:
8771:
8768:
8765:
8762:
8759:
8756:
8745:quadratic form
8732:
8727:
8700:
8697:
8674:
8671:
8668:
8663:
8656:
8653:
8650:
8636:
8618:
8615:
8612:
8609:
8597:
8594:
8581:
8578:
8575:
8572:
8569:
8548:
8545:
8542:
8539:
8536:
8515:
8508:
8504:
8498:
8493:
8490:
8487:
8482:
8478:
8471:
8467:
8462:
8440:
8437:
8434:
8425:of an element
8410:
8405:
8400:
8395:
8391:
8387:
8365:
8341:
8338:
8318:
8298:
8293:
8289:
8265:
8241:
8226:Main article:
8223:
8220:
8198:
8193:
8162:
8157:
8133:
8128:
8084:
8064:
8044:
8041:
8038:
8035:
8032:
8012:
8009:
7989:
7961:
7956:
7951:
7925:
7921:
7916:
7911:
7907:
7903:
7899:
7896:
7892:
7887:
7883:
7879:
7875:
7872:
7869:
7864:
7859:
7854:
7850:
7846:
7841:
7836:
7832:
7827:
7823:
7819:
7815:
7812:
7809:
7806:
7803:
7781:
7776:
7762:
7759:
7747:
7743:
7740:
7737:
7715:
7711:
7697:space gives a
7684:
7680:
7650:
7645:
7642:
7639:
7636:
7633:
7628:
7625:
7622:
7619:
7593:
7587:
7583:
7560:
7556:
7531:
7528:
7525:
7522:
7496:
7491:
7486:
7483:
7459:
7455:
7451:
7445:
7439:
7435:
7426:
7421:
7416:
7413:
7410:
7407:
7403:
7397:
7393:
7387:
7382:
7377:
7374:
7371:
7367:
7363:
7360:
7347:
7343:
7339:
7333:
7325:
7320:
7315:
7311:
7307:
7299:
7295:
7292:
7288:
7282:
7277:
7272:
7268:
7264:
7261:
7239:
7235:
7232:
7229:
7205:
7201:
7178:
7174:
7160:
7157:
7149:Lebesgue space
7134:
7130:
7097:
7077:
7057:
7054:
7034:
6962:
6959:
6942:
6939:
6936:
6933:
6930:
6927:
6924:
6921:
6918:
6915:
6912:
6892:
6889:
6869:
6866:
6863:
6839:
6835:
6830:
6826:
6822:
6819:
6816:
6812:
6806:
6802:
6796:
6793:
6789:
6782:
6778:
6774:
6771:
6766:
6762:
6758:
6738:
6735:
6723:
6720:
6717:
6693:
6690:
6668:
6664:
6659:
6654:
6650:
6646:
6642:
6639:
6636:
6632:
6627:
6623:
6619:
6614:
6610:
6607:
6602:
6598:
6593:
6589:
6569:
6566:
6561:
6557:
6553:
6550:
6547:
6542:
6538:
6534:
6529:
6525:
6521:
6518:
6514:
6492:
6476:Main article:
6462:
6459:
6454:
6450:
6446:
6443:
6428:
6425:
6413:
6405:
6401:
6396:
6392:
6387:
6381:
6376:
6372:
6367:
6363:
6356:
6352:
6348:
6326:
6318:
6314:
6309:
6305:
6299:
6295:
6289:
6284:
6280:
6275:
6271:
6263:
6259:
6255:
6251:
6229:
6226:
6223:
6220:
6196:
6192:
6188:
6163:
6143:
6135:
6132:
6129:
6124:
6120:
6115:
6111:
6104:
6101:
6098:
6093:
6087:
6082:
6078:
6074:
6067:
6060:
6056:
6049:
6045:
6040:
6036:
6033:
6021:therefore, is
6010:
6007:
5985:
5977:
5974:
5971:
5966:
5962:
5957:
5953:
5946:
5943:
5940:
5935:
5930:
5926:
5922:
5915:
5911:
5904:
5899:
5895:
5890:
5886:
5878:
5874:
5870:
5866:
5844:
5805:
5802:
5799:
5794:
5790:
5769:
5749:
5745:
5736:
5731:
5726:
5723:
5720:
5717:
5714:
5711:
5708:
5705:
5702:
5698:
5692:
5688:
5665:
5661:
5640:
5637:
5634:
5631:
5628:
5625:
5601:
5598:
5595:
5592:
5589:
5586:
5564:
5561:
5557:
5552:
5549:
5546:
5543:
5538:
5534:
5531:
5528:
5525:
5517:
5513:
5509:
5502:
5498:
5493:
5489:
5486:
5483:
5480:
5456:
5453:
5450:
5447:
5444:
5441:
5438:
5435:
5412:
5409:
5406:
5403:
5400:
5395:
5391:
5368:
5364:
5358:
5353:
5349:
5341:
5337:
5330:
5320:
5316:
5311:
5305:
5300:
5295:
5291:
5287:
5282:
5277:
5272:
5267:
5263:
5259:
5254:
5226:
5221:
5217:
5192:
5188:
5165:
5161:
5157:
5153:
5149:
5144:
5139:
5135:
5130:
5126:
5106:
5103:
5100:
5096:
5093:
5089:
5064:
5060:
5055:
5051:
5031:
5028:
5025:
5022:
4995:
4975:
4971:
4966:
4962:
4958:
4952:
4948:
4944:
4939:
4935:
4930:
4926:
4898:
4878:
4858:
4847:Euclidean norm
4834:
4831:
4828:
4804:
4801:
4798:
4795:
4775:
4770:
4766:
4762:
4757:
4751:
4746:
4741:
4737:
4733:
4726:
4721:
4718:
4715:
4711:
4706:
4701:
4696:
4692:
4687:
4683:
4663:
4658:
4654:
4650:
4647:
4644:
4639:
4635:
4631:
4628:
4624:
4601:
4597:
4576:
4556:
4553:
4550:
4535:Main article:
4532:
4526:
4511:
4507:
4501:
4496:
4493:
4490:
4486:
4455:
4451:
4421:
4417:
4404:cross polytope
4389:
4386:
4354:
4350:
4345:
4341:
4337:
4331:
4326:
4323:
4320:
4316:
4312:
4307:
4303:
4298:
4294:
4280:Main article:
4277:
4274:
4262:
4256:
4252:
4248:
4242:
4239:
4235:
4231:
4194:
4189:
4164:
4157:
4149:
4145:
4140:
4134:
4130:
4123:
4119:
4115:
4114:
4112:
4085:
4064:
4058:
4049:
4044:
4037:
4034:
4030:
4026:
3996:
3989:
3982:
3979:
3970:
3966:
3962:
3959:
3956:
3951:
3944:
3941:
3932:
3928:
3922:
3915:
3910:
3905:
3901:
3897:
3892:
3889:
3886:
3881:
3876:
3871:
3867:
3863:
3856:
3853:
3849:
3845:
3825:
3820:
3815:
3790:
3778:
3775:
3763:
3758:
3753:
3727:
3722:
3717:
3694:
3673:
3669:
3647:
3642:
3639:
3635:
3630:
3627:
3623:
3618:
3615:
3612:
3609:
3606:
3579:
3575:
3571:
3566:
3562:
3558:
3553:
3549:
3545:
3540:
3536:
3529:
3521:
3516:
3512:
3505:
3495:
3491:
3487:
3481:
3478:
3475:
3472:
3463:is defined by
3447:
3432:absolute value
3418:
3396:
3375:
3372:
3364:
3361:
3358:
3355:
3352:
3349:
3329:
3325:
3301:
3297:
3273:
3269:
3248:
3244:
3213:
3210:
3198:
3194:
3167:
3164:
3139:
3133:
3130:
3105:
3102:
3099:
3096:
3093:
3090:
3066:
3062:
3058:
3053:
3049:
3026:
3023:
3020:
3017:
2997:
2992:
2987:
2963:absolute value
2959:complex number
2948:
2945:
2930:
2907:
2904:
2901:
2896:
2867:
2863:
2837:
2833:
2799:
2795:
2761:
2757:
2731:
2727:
2714:quadratic norm
2699:
2693:
2689:
2685:
2679:
2676:
2672:
2668:
2625:
2620:
2615:
2573:Euclidean norm
2558:
2551:
2546:
2542:
2538:
2535:
2532:
2527:
2522:
2518:
2512:
2507:
2503:
2498:
2494:
2473:
2467:
2463:
2459:
2456:
2453:
2448:
2444:
2440:
2435:
2431:
2426:
2422:
2418:
2397:
2392:
2387:
2362:
2345:Euclidean norm
2340:
2339:Euclidean norm
2337:
2324:
2321:
2300:
2296:
2293:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2245:
2241:
2237:
2216:
2212:
2190:
2168:
2147:
2144:
2141:
2137:
2133:
2130:
2106:
2086:
2050:
2046:
2042:
2031:absolute value
2026:
2023:
2010:
2007:
1986:
1981:
1977:
1973:
1967:
1964:
1961:
1957:
1936:
1914:
1910:
1889:
1886:
1881:
1877:
1871:
1867:
1861:
1858:
1855:
1851:
1847:
1844:
1824:
1798:
1795:
1792:
1787:
1782:
1778:
1774:
1769:
1764:
1760:
1747:
1744:
1730:
1726:
1722:
1697:
1677:
1674:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1633:
1630:
1627:
1607:
1604:
1583:
1579:
1576:
1573:
1570:
1558:
1555:
1542:
1539:
1519:
1499:
1479:
1476:
1448:
1442:
1439:
1433:
1430:
1427:
1424:
1418:
1415:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1346:
1326:
1317:The relation "
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1239:
1236:
1233:
1230:
1210:
1207:
1204:
1184:
1164:
1140:
1120:
1100:
1097:
1077:
1057:
1045:
1042:
1039:
1035:
1030:
1029:
1018:
1015:
1012:
1009:
989:
986:
983:
980:
977:
974:
963:Non-negativity
960:
945:
925:
922:
919:
916:
913:
910:
890:
865:
861:
858:
855:
852:
843:is a function
832:
817:
816:
815:
814:
803:
800:
797:
777:
774:
771:
768:
765:
762:
743:
740:
737:
734:
714:
711:
708:
705:
702:
699:
696:
674:
671:
668:
648:
645:
642:
639:
636:
633:
613:
610:
607:
604:
589:/positiveness/
584:
573:
570:
550:
547:
544:
524:
521:
518:
515:
511:
507:
503:
499:
496:
493:
490:
487:
484:
470:
459:
456:
453:
450:
447:
444:
424:
421:
418:
415:
412:
409:
406:
403:
400:
397:
394:
391:
388:
385:
382:
379:
348:
337:absolute value
323:
319:
315:
293:
289:
286:
283:
280:
257:
233:
229:
208:
185:
170:
167:
149:
95:Euclidean norm
26:
9:
6:
4:
3:
2:
15185:
15174:
15171:
15169:
15166:
15164:
15161:
15160:
15158:
15143:
15135:
15134:
15131:
15125:
15122:
15120:
15117:
15115:
15112:
15110:
15106:
15102:
15100:) convex
15099:
15096:
15094:
15091:
15089:
15085:
15083:
15080:
15078:
15075:
15073:
15072:Semi-complete
15070:
15068:
15065:
15063:
15060:
15058:
15054:
15051:
15049:
15045:
15043:
15040:
15038:
15035:
15033:
15030:
15028:
15025:
15023:
15020:
15018:
15015:
15013:
15010:
15008:
15005:
15003:
15000:
14998:
14995:
14993:
14990:
14988:
14987:Infrabarreled
14985:
14983:
14980:
14978:
14975:
14971:
14968:
14967:
14966:
14963:
14961:
14958:
14956:
14953:
14951:
14948:
14946:
14945:Distinguished
14943:
14941:
14938:
14936:
14933:
14931:
14928:
14926:
14923:
14921:
14917:
14913:
14911:
14908:
14906:
14902:
14898:
14896:
14893:
14891:
14888:
14886:
14883:
14882:
14880:
14878:Types of TVSs
14876:
14870:
14866:
14862:
14860:
14857:
14855:
14852:
14850:
14847:
14845:
14842:
14840:
14836:
14832:
14830:
14827:
14826:
14824:
14820:
14814:
14811:
14809:
14806:
14804:
14801:
14799:
14798:Prevalent/Shy
14796:
14794:
14791:
14789:
14788:Extreme point
14786:
14784:
14778:
14776:
14770:
14768:
14765:
14763:
14760:
14758:
14755:
14753:
14750:
14748:
14745:
14743:
14740:
14738:
14735:
14733:
14730:
14728:
14725:
14724:
14722:
14720:Types of sets
14718:
14712:
14709:
14707:
14704:
14702:
14699:
14697:
14694:
14690:
14687:
14685:
14682:
14680:
14677:
14676:
14675:
14672:
14670:
14667:
14663:
14662:Discontinuous
14660:
14658:
14655:
14653:
14650:
14648:
14645:
14643:
14640:
14638:
14635:
14633:
14630:
14629:
14628:
14625:
14621:
14618:
14617:
14616:
14613:
14612:
14610:
14606:
14600:
14597:
14593:
14590:
14589:
14588:
14585:
14582:
14579:
14577:
14573:
14570:
14568:
14565:
14563:
14560:
14558:
14555:
14553:
14550:
14549:
14547:
14545:
14541:
14535:
14532:
14530:
14527:
14525:
14522:
14520:
14519:Metrizability
14517:
14515:
14512:
14510:
14507:
14505:
14504:Fréchet space
14502:
14500:
14497:
14495:
14492:
14490:
14487:
14485:
14482:
14481:
14479:
14475:
14470:
14463:
14458:
14456:
14451:
14449:
14444:
14443:
14440:
14428:
14420:
14419:
14416:
14410:
14407:
14405:
14402:
14400:
14399:Weak topology
14397:
14395:
14392:
14390:
14387:
14385:
14382:
14381:
14379:
14375:
14368:
14364:
14361:
14359:
14356:
14354:
14351:
14349:
14346:
14344:
14341:
14339:
14336:
14334:
14331:
14329:
14326:
14324:
14323:Index theorem
14321:
14319:
14316:
14314:
14311:
14309:
14306:
14305:
14303:
14299:
14293:
14290:
14288:
14285:
14284:
14282:
14280:Open problems
14278:
14272:
14269:
14267:
14264:
14262:
14259:
14257:
14254:
14252:
14249:
14247:
14244:
14243:
14241:
14237:
14231:
14228:
14226:
14223:
14221:
14218:
14216:
14213:
14211:
14208:
14206:
14203:
14201:
14198:
14196:
14193:
14191:
14188:
14186:
14183:
14182:
14180:
14176:
14170:
14167:
14165:
14162:
14160:
14157:
14155:
14152:
14150:
14147:
14145:
14142:
14140:
14137:
14135:
14132:
14130:
14127:
14126:
14124:
14122:
14118:
14108:
14105:
14103:
14100:
14098:
14095:
14092:
14088:
14084:
14081:
14079:
14076:
14074:
14071:
14070:
14068:
14064:
14058:
14055:
14053:
14050:
14048:
14045:
14043:
14040:
14038:
14035:
14033:
14030:
14028:
14025:
14023:
14020:
14018:
14015:
14013:
14010:
14009:
14006:
14003:
13999:
13994:
13990:
13986:
13979:
13974:
13972:
13967:
13965:
13960:
13959:
13956:
13944:
13941:
13939:
13936:
13934:
13931:
13929:
13926:
13924:
13921:
13920:
13918:
13914:
13908:
13890:
13886:
13882:
13879:
13873:
13864:
13862:
13859:
13855:
13852:
13851:
13850:
13847:
13845:
13842:
13840:
13839:
13834:
13832:
13818:
13813:
13803:
13799:
13790:
13786:
13783:
13781:
13762:
13753:
13752:
13751:
13750:
13746:
13742:
13723:
13714:
13713:
13712:
13711:
13707:
13705:
13681:
13678:
13675:
13671:
13661:
13659:
13658:
13653:
13651:
13648:
13646:
13644:
13640:
13635:
13633:
13630:
13628:
13627:
13622:
13620:
13617:
13615:
13589:
13584:
13581:
13578:
13574:
13564:
13562:
13561:
13556:
13554:
13551:
13549:
13527:
13524:
13516:
13514:
13513:
13508:
13507:
13505:
13503:
13499:
13493:
13489:
13485:
13483:
13480:
13478:
13475:
13473:
13470:
13468:
13465:
13463:
13462:Extreme point
13460:
13458:
13455:
13453:
13450:
13448:
13444:
13440:
13438:
13435:
13434:
13432:
13428:
13422:
13419:
13417:
13414:
13412:
13409:
13407:
13404:
13402:
13396:
13393:
13389:
13385:
13382:
13380:
13374:
13372:
13369:
13367:
13364:
13362:
13359:
13357:
13354:
13352:
13349:
13347:
13344:
13343:
13341:
13339:Types of sets
13337:
13330:
13326:
13323:
13319:
13316:
13314:
13311:
13309:
13306:
13305:
13304:
13301:
13297:
13294:
13292:
13289:
13287:
13284:
13283:
13282:
13279:
13275:
13272:
13270:
13267:
13265:
13262:
13260:
13257:
13255:
13252:
13250:
13247:
13246:
13245:
13242:
13238:
13235:
13233:
13230:
13228:
13225:
13223:
13220:
13218:
13215:
13214:
13213:
13210:
13208:
13205:
13203:
13202:Convex series
13200:
13198:
13197:Bochner space
13195:
13191:
13188:
13187:
13186:
13183:
13181:
13178:
13177:
13175:
13171:
13165:
13162:
13160:
13157:
13155:
13152:
13150:
13149:Riesz's lemma
13147:
13145:
13142:
13140:
13137:
13135:
13134:Mazur's lemma
13132:
13130:
13127:
13125:
13122:
13120:
13117:
13115:
13112:
13108:
13105:
13104:
13103:
13100:
13098:
13095:
13093:
13090:
13088:
13087:Gelfand–Mazur
13085:
13083:
13080:
13078:
13075:
13073:
13070:
13068:
13065:
13063:
13060:
13058:
13055:
13053:
13050:
13048:
13045:
13043:
13040:
13038:
13035:
13033:
13030:
13028:
13025:
13024:
13022:
13020:
13016:
13010:
13007:
13005:
13002:
12998:
12995:
12993:
12990:
12989:
12988:
12985:
12981:
12978:
12976:
12973:
12972:
12971:
12968:
12966:
12963:
12961:
12958:
12956:
12953:
12952:
12950:
12948:
12944:
12938:
12935:
12933:
12930:
12928:
12925:
12923:
12920:
12918:
12915:
12913:
12910:
12908:
12905:
12903:
12900:
12896:
12893:
12892:
12891:
12888:
12886:
12883:
12879:
12876:
12874:
12871:
12870:
12868:
12866:
12863:
12861:
12857:
12853:
12849:
12846:
12845:
12844:
12841:
12839:
12836:
12834:
12830:
12826:
12822:
12819:
12817:
12814:
12812:
12809:
12808:
12807:
12804:
12802:
12799:
12798:
12796:
12794:
12790:
12784:
12781:
12779:
12776:
12772:
12769:
12767:
12764:
12763:
12762:
12759:
12755:
12752:
12750:
12747:
12746:
12745:
12742:
12740:
12737:
12735:
12732:
12728:
12725:
12724:
12723:
12720:
12718:
12715:
12713:
12710:
12709:
12707:
12703:
12697:
12694:
12692:
12689:
12685:
12682:
12681:
12680:
12677:
12675:
12672:
12670:
12667:
12663:
12659:
12656:
12655:
12654:
12651:
12647:
12644:
12643:
12642:
12639:
12637:
12634:
12632:
12629:
12627:
12624:
12623:
12621:
12617:
12610:
12606:
12602:
12599:
12597:
12593:
12591:
12588:
12586:) convex
12585:
12582:
12580:
12577:
12575:
12571:
12569:
12566:
12564:
12561:
12559:
12555:
12551:
12547:
12544:
12542:
12539:
12538:
12537:
12534:
12532:
12531:Grothendieck
12529:
12527:
12524:
12520:
12517:
12516:
12515:
12512:
12510:
12507:
12506:
12504:
12500:
12495:
12488:
12483:
12481:
12476:
12474:
12469:
12468:
12465:
12457:
12453:
12449:
12443:
12439:
12435:
12431:
12427:
12423:
12419:
12413:
12409:
12405:
12401:
12397:
12393:
12389:
12383:
12379:
12375:
12371:
12367:
12363:
12359:
12355:
12349:
12345:
12340:
12336:
12332:
12328:
12322:
12318:
12314:
12309:
12305:
12301:
12297:
12291:
12287:
12283:
12279:
12274:
12270:
12266:
12262:
12260:3-540-13627-4
12256:
12252:
12248:
12244:
12240:
12239:
12226:
12222:
12216:
12209:
12204:
12196:
12194:0-8018-5413-X
12190:
12186:
12182:
12176:
12174:
12166:
12161:
12153:
12151:0-387-95385-X
12147:
12143:
12136:
12129:
12125:
12121:
12117:
12113:
12111:90-277-2186-6
12107:
12103:
12099:
12095:
12088:
12069:
12064:
12037:
12032:
12015:
12001:
11997:
11990:
11982:
11976:
11972:
11965:
11951:
11947:
11946:"Vector Norm"
11940:
11938:
11936:
11928:
11927:Wilansky 2013
11923:
11908:
11901:
11894:
11892:
11890:
11882:
11877:
11870:. p. 25.
11869:
11862:
11855:
11854:Kubrusly 2011
11850:
11848:
11839:
11835:
11828:
11822:
11818:
11814:
11807:
11799:
11795:
11791:
11787:
11783:
11779:
11775:
11768:
11753:
11749:
11743:
11729:
11725:
11719:
11711:
11705:
11701:
11698:
11691:
11689:
11684:
11673:
11670:
11664:
11661:
11655:
11652:
11646:
11643:
11640:
11639:Operator norm
11637:
11634:
11631:
11628:
11625:
11622:
11619:
11616:
11613:
11610:
11607:
11604:
11601:
11595:
11592:
11590:
11587:
11581:
11578:
11575:
11572:
11571:
11547:
11544:
11539:
11535:
11528:
11525:
11503:
11499:
11478:
11470:
11467:
11463:
11459:
11458:
11440:
11436:
11432:
11429:
11421:
11405:
11397:
11378:
11375:
11372:
11366:
11363:
11343:
11335:
11316:
11293:
11290:
11267:
11256:
11255:
11254:
11251:
11249:
11244:
11242:
11238:
11219:
11215:
11211:
11208:
11205:
11194:
11178:
11171:of seminorms
11155:
11144:
11140:
11135:
11121:
11117:
11113:
11110:
11104:
11096:
11092:
11088:
11085:
11082:
11079:
11075:
11068:
11062:
11056:
11049:
11045:
11042:
11036:
11028:
11024:
11020:
11017:
11014:
11011:
11007:
10998:
10955:
10952:
10949:
10946:
10943:
10940:
10937:
10934:
10931:
10923:
10920:
10911:
10905:
10897:
10893:
10872:
10869:
10861:
10860:
10841:
10837:
10816:
10813:
10793:
10786:
10783:
10767:
10757:
10747:
10745:
10739:
10725:
10712:
10706:
10703:
10698:
10690:
10682:
10677:
10672:
10664:
10658:
10653:
10645:
10639:
10626:
10603:
10590:
10584:
10581:
10576:
10568:
10562:
10549:
10517:
10509:
10504:
10499:
10491:
10485:
10472:
10448:
10440:
10432:
10427:
10422:
10414:
10408:
10403:
10395:
10383:
10370:
10365:
10357:
10346:
10342:
10338:
10335:
10332:
10328:
10324:
10317:
10313:
10308:
10300:
10294:
10289:
10281:
10258:
10253:
10226:
10223:
10220:
10217:
10214:
10194:
10189:
10181:
10175:
10172:
10167:
10159:
10153:
10148:
10140:
10134:
10115:
10112:
10109:
10089:
10069:
10061:
10041:
10019:
10011:
9986:
9978:
9966:
9953:
9947:
9944:
9941:
9935:
9932:
9924:
9921:
9918:
9915:
9912:
9901:
9898:
9895:
9889:
9883:
9880:
9877:
9871:
9865:
9862:
9859:
9830:
9824:
9821:
9810:
9794:
9785:
9765:
9754:
9751:
9746:
9742:
9716:
9713:
9705:
9684:
9680:
9668:
9663:
9659:
9654:
9640:
9620:
9617:
9614:
9606:
9602:
9586:
9578:
9574:
9570:
9560:
9558:
9554:
9550:
9546:
9542:
9534:
9529:
9525:
9512:
9507:
9499:
9491:
9483:
9477:
9473:
9466:
9463:
9460:
9453:
9444:
9428:
9425:
9420:
9417:
9412:
9407:
9404:
9396:
9388:
9380:
9372:
9366:
9355:
9352:
9349:
9334:
9330:
9314:
9310:
9299:
9285:
9277:
9261:
9241:
9235:
9232:
9229:
9209:
9206:
9203:
9200:
9197:
9194:
9178:
9172:
9169:
9163:
9157:
9149:
9143:
9140:
9137:
9131:
9123:
9107:
9104:
9076:
9073:
9070:
9063:For any norm
9056:
9054:
9050:
9046:
9008:
8984:
8960:
8948:
8932:
8926:
8920:
8914:
8908:
8905:
8899:
8896:
8890:
8870:
8850:
8830:
8827:
8807:
8799:
8794:
8792:
8777:
8773:
8769:
8766:
8760:
8754:
8746:
8730:
8725:
8714:
8698:
8695:
8688:
8669:
8666:
8661:
8654:
8651:
8640:
8634:
8632:
8613:
8607:
8593:
8576:
8573:
8570:
8559:
8543:
8540:
8537:
8513:
8506:
8502:
8496:
8488:
8480:
8476:
8469:
8465:
8460:
8451:is the value
8438:
8435:
8432:
8424:
8408:
8403:
8398:
8393:
8389:
8385:
8363:
8355:
8339:
8336:
8316:
8296:
8291:
8287:
8279:
8263:
8255:
8239:
8229:
8219:
8217:
8212:
8196:
8180:
8178:
8160:
8131:
8116:
8111:
8109:
8105:
8100:
8098:
8097:parallelogram
8082:
8062:
8042:
8036:
8033:
8010:
8007:
7987:
7980:
7977:
7972:
7959:
7954:
7940:is a norm on
7923:
7914:
7909:
7905:
7901:
7897:
7894:
7890:
7885:
7881:
7877:
7867:
7862:
7857:
7852:
7848:
7844:
7839:
7834:
7830:
7825:
7821:
7817:
7813:
7810:
7804:
7779:
7758:
7745:
7741:
7738:
7735:
7713:
7709:
7700:
7682:
7678:
7668:
7666:
7661:
7648:
7640:
7637:
7634:
7626:
7620:
7609:
7608:inner product
7604:
7591:
7581:
7554:
7545:
7544:supremum norm
7526:
7520:
7512:
7494:
7484:
7481:
7472:
7457:
7453:
7449:
7437:
7424:
7411:
7405:
7395:
7391:
7380:
7375:
7372:
7369:
7361:
7345:
7341:
7337:
7323:
7318:
7313:
7309:
7305:
7293:
7290:
7286:
7275:
7270:
7262:
7251:
7237:
7233:
7230:
7227:
7219:
7203:
7199:
7176:
7172:
7156:
7154:
7150:
7132:
7128:
7119:
7115:
7111:
7095:
7075:
7055:
7052:
7032:
7024:
7021:
7018:
7015:
7011:
7007:
7003:
6998:
6996:
6992:
6988:
6987:
6982:
6978:
6972:
6968:
6958:
6956:
6940:
6934:
6931:
6928:
6922:
6919:
6913:
6890:
6887:
6864:
6853:
6837:
6828:
6824:
6820:
6817:
6810:
6804:
6800:
6794:
6791:
6787:
6780:
6776:
6764:
6760:
6748:
6744:
6734:
6721:
6718:
6715:
6707:
6691:
6688:
6679:
6666:
6662:
6657:
6652:
6648:
6644:
6640:
6637:
6634:
6630:
6625:
6621:
6617:
6612:
6605:
6567:
6559:
6555:
6551:
6548:
6545:
6540:
6536:
6532:
6527:
6523:
6516:
6479:
6460:
6457:
6444:
6433:
6424:
6411:
6403:
6379:
6374:
6324:
6316:
6297:
6293:
6287:
6282:
6261:
6257:
6240:this becomes
6227:
6224:
6221:
6218:
6209:
6190:
6177:
6161:
6141:
6133:
6130:
6127:
6122:
6102:
6099:
6096:
6076:
6065:
6047:
6008:
6005:
5996:
5983:
5975:
5972:
5969:
5964:
5944:
5941:
5938:
5933:
5928:
5924:
5920:
5913:
5909:
5902:
5897:
5876:
5872:
5842:
5833:
5831:
5827:
5823:
5819:
5800:
5792:
5788:
5767:
5747:
5734:
5721:
5715:
5712:
5706:
5700:
5690:
5686:
5663:
5659:
5638:
5635:
5632:
5629:
5626:
5623:
5615:
5599:
5596:
5593:
5590:
5587:
5584:
5575:
5562:
5559:
5547:
5541:
5529:
5523:
5515:
5511:
5507:
5500:
5496:
5487:
5484:
5481:
5470:
5454:
5448:
5445:
5439:
5436:
5426:
5425:measure space
5407:
5404:
5401:
5393:
5389:
5366:
5362:
5351:
5347:
5339:
5335:
5328:
5318:
5314:
5303:
5298:
5293:
5289:
5285:
5280:
5275:
5270:
5265:
5261:
5257:
5244:
5224:
5219:
5215:
5206:
5190:
5155:
5142:
5137:
5117:meaning that
5104:
5098:
5094:
5091:
5080:
5079:inner product
5062:
5053:
5029:
5026:
5023:
5020:
5011:
5009:
4993:
4973:
4969:
4964:
4960:
4956:
4950:
4942:
4916:
4912:
4911:infinity norm
4896:
4856:
4848:
4832:
4829:
4826:
4818:
4802:
4799:
4796:
4793:
4773:
4768:
4764:
4760:
4755:
4749:
4744:
4739:
4735:
4731:
4724:
4719:
4716:
4713:
4709:
4704:
4699:
4694:
4656:
4652:
4648:
4645:
4642:
4637:
4633:
4626:
4599:
4595:
4574:
4554:
4551:
4548:
4538:
4530:
4525:
4509:
4505:
4499:
4494:
4491:
4488:
4484:
4476:In contrast,
4474:
4471:
4469:
4453:
4449:
4439:
4435:
4419:
4415:
4405:
4400:
4387:
4384:
4376:
4372:
4368:
4352:
4348:
4343:
4339:
4335:
4329:
4324:
4321:
4318:
4314:
4310:
4305:
4283:
4273:
4260:
4250:
4240:
4221:
4217:
4212:
4210:
4192:
4155:
4147:
4143:
4138:
4132:
4128:
4121:
4117:
4110:
4100:
4099:column vector
4062:
4047:
4035:
4016:
4015:inner product
4012:
4007:
3994:
3987:
3977:
3968:
3964:
3960:
3957:
3954:
3949:
3939:
3930:
3926:
3920:
3913:
3908:
3903:
3899:
3895:
3890:
3887:
3884:
3879:
3874:
3869:
3865:
3861:
3854:
3823:
3818:
3804:
3803:complex space
3801:-dimensional
3788:
3774:
3761:
3756:
3741:
3725:
3720:
3671:
3640:
3637:
3628:
3625:
3616:
3613:
3610:
3607:
3604:
3577:
3573:
3569:
3564:
3560:
3556:
3551:
3547:
3543:
3538:
3534:
3527:
3519:
3514:
3510:
3503:
3493:
3489:
3485:
3479:
3473:
3462:
3435:
3433:
3373:
3370:
3362:
3359:
3356:
3353:
3350:
3347:
3327:
3315:
3299:
3287:
3271:
3246:
3233:
3229:
3223:
3219:
3209:
3196:
3192:
3184:
3162:
3137:
3128:
3103:
3100:
3097:
3094:
3091:
3088:
3064:
3060:
3056:
3051:
3047:
3024:
3021:
3018:
3015:
2995:
2990:
2976:
2972:
2971:complex plane
2968:
2964:
2960:
2954:
2944:
2942:
2928:
2905:
2902:
2899:
2883:
2881:
2865:
2861:
2851:
2835:
2831:
2821:
2817:
2813:
2797:
2793:
2783:
2779:
2775:
2759:
2755:
2745:
2729:
2725:
2715:
2710:
2697:
2687:
2677:
2658:
2654:
2650:
2646:
2642:
2641:inner product
2637:
2623:
2618:
2602:
2600:
2596:
2592:
2588:
2584:
2580:
2579:
2574:
2569:
2556:
2549:
2544:
2540:
2536:
2533:
2530:
2525:
2520:
2516:
2510:
2505:
2471:
2465:
2461:
2457:
2454:
2451:
2446:
2442:
2438:
2433:
2429:
2424:
2420:
2395:
2390:
2376:
2373:-dimensional
2360:
2350:
2346:
2336:
2322:
2319:
2294:
2291:
2271:
2262:
2256:
2250:
2247:
2239:
2214:
2145:
2142:
2131:
2128:
2120:
2104:
2084:
2075:
2073:
2069:
2065:
2044:
2032:
2022:
2008:
2005:
1998:is a norm on
1984:
1979:
1975:
1971:
1965:
1962:
1959:
1955:
1934:
1912:
1908:
1887:
1884:
1879:
1875:
1869:
1865:
1859:
1856:
1853:
1849:
1845:
1842:
1822:
1814:
1796:
1793:
1790:
1785:
1780:
1776:
1772:
1767:
1762:
1758:
1743:
1724:
1711:
1695:
1675:
1669:
1663:
1660:
1654:
1631:
1628:
1625:
1605:
1602:
1574:
1571:
1568:
1554:
1540:
1537:
1517:
1497:
1477:
1474:
1466:
1462:
1446:
1440:
1437:
1431:
1428:
1425:
1422:
1416:
1413:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1364:
1360:
1344:
1324:
1304:
1298:
1292:
1289:
1286:
1280:
1274:
1271:
1265:
1259:
1256:
1237:
1234:
1231:
1228:
1208:
1205:
1202:
1182:
1162:
1154:
1138:
1118:
1098:
1095:
1075:
1055:
1048:Suppose that
1041:
1037:
1033:
1016:
1013:
1010:
1007:
987:
984:
978:
972:
964:
959:
958:
957:
943:
923:
920:
914:
908:
888:
880:
856:
853:
850:
830:
822:
801:
798:
795:
775:
772:
766:
760:
741:
738:
735:
732:
712:
709:
706:
700:
694:
686:
685:
672:
669:
666:
646:
643:
637:
631:
611:
608:
605:
602:
588:
585:
571:
568:
548:
545:
542:
519:
513:
505:
497:
491:
488:
482:
474:
471:
457:
454:
451:
448:
445:
442:
419:
413:
410:
404:
398:
395:
389:
386:
383:
377:
369:
365:
364:Subadditivity
362:
361:
360:
346:
338:
317:
284:
281:
278:
271:
255:
247:
231:
206:
199:
183:
176:
166:
164:
147:
138:
133:
131:
127:
123:
118:
116:
115:inner product
112:
108:
104:
100:
96:
93:, called the
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
49:
45:
41:
37:
33:
19:
15048:Polynomially
14977:Grothendieck
14970:tame Fréchet
14920:Bornological
14780:Linear cone
14772:Convex cone
14747:Banach disks
14695:
14689:Sesquilinear
14544:Main results
14534:Vector space
14489:Completeness
14484:Banach space
14389:Balanced set
14363:Distribution
14301:Applications
14154:Krein–Milman
14139:Closed graph
13916:Applications
13837:
13748:
13709:
13656:
13642:
13638:
13625:
13559:
13511:
13398:Linear cone
13391:
13387:
13376:Convex cone
13269:Paley–Wiener
13129:Mackey–Arens
13119:Krein–Milman
13072:Closed range
13067:Closed graph
13037:Banach–Mazur
12917:Self-adjoint
12821:sesquilinear
12683:
12554:Polynomially
12494:Banach space
12437:
12407:
12373:
12343:
12312:
12277:
12246:
12236:Bibliography
12224:
12215:
12203:
12184:
12160:
12141:
12135:
12093:
12087:
12014:
12003:. Retrieved
11999:
11989:
11970:
11964:
11953:. Retrieved
11949:
11922:
11912:September 7,
11910:. Retrieved
11906:
11876:
11867:
11861:
11833:
11812:
11806:
11781:
11777:
11767:
11756:. Retrieved
11751:
11748:"Pseudonorm"
11742:
11731:. Retrieved
11727:
11718:
11696:
11468:. Precisely:
11394:is its open
11252:
11245:
11136:
11134:Conversely:
10857:
10759:
10740:
10384:
10055:
9967:
9655:
9601:superellipse
9566:
9538:
9533:unit circles
9300:
9062:
8800:property of
8798:homomorphism
8795:
8639:null vectors
8599:
8422:
8231:
8216:matrix norms
8213:
8181:
8112:
8101:
8055:In 2D, with
7973:
7764:
7669:
7665:Banach space
7662:
7605:
7511:Haar measure
7473:
7252:
7162:
7109:
7022:
7019:
7016:
7013:
7010:David Donoho
6999:
6984:
6974:
6954:
6851:
6745:and for the
6740:
6680:
6481:
6478:Maximum norm
6210:
5997:
5834:
5576:
5238:
5012:
4915:maximum norm
4817:taxicab norm
4540:
4528:
4475:
4472:
4441:
4407:
4401:
4285:
4213:
4207:denotes its
4008:
3780:
3436:
3232:real numbers
3225:
2966:
2956:
2884:
2853:
2823:
2819:
2781:
2777:
2747:
2717:
2713:
2711:
2638:
2603:
2598:
2594:
2590:
2586:
2577:
2576:
2572:
2571:This is the
2570:
2352:
2076:
2028:
1749:
1560:
1152:
1047:
1031:
818:
339:of a scalar
245:
175:vector space
172:
163:directed set
136:
134:
129:
119:
106:
102:
67:vector space
58:
52:
44:Norm (group)
15042:Quasinormed
14955:FK-AK space
14849:Linear span
14844:Convex hull
14829:Affine hull
14632:Almost open
14572:Hahn–Banach
14318:Heat kernel
14308:Hardy space
14215:Trace class
14129:Hahn–Banach
14091:Topological
13637:Continuous
13472:Linear span
13457:Convex hull
13437:Affine hull
13296:holomorphic
13232:holomorphic
13212:Derivatives
13102:Hahn–Banach
13042:Banach–Saks
12960:C*-algebras
12927:Trace class
12890:Functionals
12778:Ultrastrong
12691:Quasinormed
12181:Golub, Gene
12165:Trèves 2006
11754:(in German)
11621:Matrix norm
11589:Gowers norm
11253:norm case:
11143:local basis
10885:defined as
10856:called the
10054:are called
9706:in norm to
9702:is said to
9669:of vectors
9569:unit circle
9563:Equivalence
9298:are equal.
8256:of a field
8104:octahedrons
7728:space when
7542:, giving a
7250:with norms
5245:defined by
4869:approaches
4845:we get the
4815:we get the
4373:borough of
4367:street grid
4011:square root
3461:quaternions
3286:quaternions
2818:called the
2782:square norm
2649:dot product
2593:oot of the
2119:isomorphism
1813:Hamel basis
1151:are called
111:square root
55:mathematics
18:Vector norm
15157:Categories
15082:Stereotype
14940:(DF)-space
14935:Convenient
14674:Functional
14642:Continuous
14627:Linear map
14567:F. Riesz's
14509:Linear map
14251:C*-algebra
14066:Properties
13390:), and (Hw
13291:continuous
13227:functional
12975:C*-algebra
12860:Continuous
12722:Dual space
12696:Stereotype
12674:Metrizable
12601:Projective
12317:Birkhäuser
12018:Except in
12005:2020-08-24
11955:2020-08-24
11836:. p.
11758:2022-05-12
11733:2022-05-12
11679:References
11594:Kadec norm
11580:F-seminorm
11563:is a norm.
11334:separating
11241:continuous
10058:equivalent
9968:Two norms
9331:, we have
9059:Properties
8713:involution
8354:embeddings
8228:Field norm
7088:-norms as
7006:statistics
6965:See also:
6903:such that
3430:are their
3218:Quaternion
3216:See also:
2951:See also:
2180:is either
1561:If a norm
1490:The norms
1461:transitive
1153:equivalent
595:: for all
169:Definition
137:pseudonorm
36:Ideal norm
32:Field norm
15098:Uniformly
15057:Reflexive
14905:Barrelled
14901:Countably
14813:Symmetric
14711:Transpose
14225:Unbounded
14220:Transpose
14178:Operators
14107:Separable
14102:Reflexive
14087:Algebraic
14073:Barrelled
13849:Sobolev W
13792:Schwartz
13767:∞
13728:∞
13724:ℓ
13690:Ω
13676:λ
13534:Σ
13416:Symmetric
13351:Absorbing
13264:regulated
13244:Integrals
13097:Goldstine
12932:Transpose
12869:Fredholm
12739:Ultraweak
12727:Dual norm
12658:Seminorms
12626:Barrelled
12596:Injective
12584:Uniformly
12558:Reflexive
12456:849801114
12426:853623322
12406:(2006) .
12396:840278135
12362:144216834
12335:710154895
12245:(1987) .
11798:0012-7094
11518:(so that
11396:unit ball
11111:≤
11083:∈
11069:⊆
11057:⊆
11015:∈
10950:∈
10924:∈
10721:∞
10717:‖
10710:‖
10704:≤
10695:‖
10688:‖
10678:≤
10669:‖
10662:‖
10659:≤
10650:‖
10643:‖
10640:≤
10635:∞
10631:‖
10624:‖
10616:That is,
10599:∞
10595:‖
10588:‖
10582:≤
10573:‖
10566:‖
10563:≤
10558:∞
10554:‖
10547:‖
10526:∞
10522:‖
10515:‖
10505:≤
10496:‖
10489:‖
10486:≤
10481:∞
10477:‖
10470:‖
10445:‖
10438:‖
10428:≤
10419:‖
10412:‖
10409:≤
10400:‖
10393:‖
10362:‖
10355:‖
10336:−
10314:≤
10305:‖
10298:‖
10295:≤
10286:‖
10279:‖
10224:≥
10190:α
10186:‖
10179:‖
10173:≤
10168:β
10164:‖
10157:‖
10154:≤
10149:α
10145:‖
10138:‖
10113:∈
10020:β
10016:‖
10012:⋅
10009:‖
9987:α
9983:‖
9979:⋅
9976:‖
9936:∈
9922:∈
9905:‖
9899:−
9893:‖
9887:‖
9881:−
9875:‖
9869:‖
9863:−
9857:‖
9834:‖
9831:⋅
9828:‖
9792:∞
9789:→
9763:→
9752:−
9662:Hausdorff
9618:≥
9504:‖
9497:‖
9488:‖
9481:‖
9478:≤
9470:⟩
9458:⟨
9393:‖
9386:‖
9377:‖
9370:‖
9367:≤
9359:⟩
9347:⟨
9276:transpose
9239:→
9204:∈
9170:−
9150:≥
9141:±
9080:→
9047:. In the
8778:∗
8726:∗
8662:∗
8507:μ
8489:α
8477:σ
8466:∏
8436:∈
8433:α
8421:then the
8390:σ
8292:μ
8040:‖
8031:‖
8023:equal to
7976:injective
7808:‖
7802:‖
7710:ℓ
7679:ℓ
7667:article.
7644:⟩
7632:⟨
7624:‖
7618:‖
7586:∞
7559:∞
7555:ℓ
7530:∞
7524:→
7485:⊆
7392:∫
7366:‖
7359:‖
7294:∈
7287:∑
7267:‖
7260:‖
7231:≥
7173:ℓ
6917:‖
6911:‖
6868:‖
6865:⋅
6862:‖
6792:−
6777:∑
6773:↦
6737:Zero norm
6706:hypercube
6638:…
6601:∞
6597:‖
6588:‖
6549:…
6453:∞
6449:‖
6442:‖
6400:‖
6391:‖
6371:‖
6362:‖
6351:∂
6347:∂
6313:‖
6304:‖
6279:‖
6270:‖
6254:∂
6250:∂
6191:⋅
6162:∘
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6110:‖
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6077:∘
6055:∂
6044:‖
6035:‖
6032:∂
5973:−
5961:‖
5952:‖
5942:−
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5885:‖
5869:∂
5865:∂
5760:(without
5748:μ
5713:−
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5537:¯
5512:∫
5492:⟩
5479:⟨
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5443:Σ
5408:μ
5357:¯
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5315:ℓ
5310:⟩
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5216:ℓ
5164:⟩
5148:⟨
5134:‖
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5102:⟩
5099:⋅
5092:⋅
5088:⟨
5059:‖
5054:⋅
5050:‖
4938:∞
4934:‖
4925:‖
4877:∞
4849:, and as
4710:∑
4691:‖
4682:‖
4646:…
4596:ℓ
4552:≥
4485:∑
4450:ℓ
4416:ℓ
4375:Manhattan
4315:∑
4302:‖
4293:‖
4251:⋅
4238:‖
4230:‖
4139:…
4033:‖
4025:‖
3981:¯
3958:⋯
3943:¯
3888:⋯
3852:‖
3844:‖
3740:octonions
3515:∗
3494:∗
3477:‖
3471:‖
3314:octonions
3230:over the
3166:¯
3132:¯
2862:ℓ
2756:ℓ
2688:⋅
2675:‖
2667:‖
2651:of their
2534:⋯
2502:‖
2493:‖
2455:…
2295:∈
2140:→
2077:Any norm
1963:∈
1956:∑
1885:∈
1857:∈
1850:∑
1794:∈
1763:∙
1658:‖
1652:‖
1629:∈
1578:→
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1426:≤
1385:≤
1379:≤
1363:symmetric
1359:reflexive
1287:≤
1272:≤
1232:∈
1011:∈
985:≥
936:and that
860:→
736:∈
606:∈
546:∈
452:∈
396:≤
288:→
148:≤
135:The term
103:magnitude
15142:Category
15093:Strictly
15067:Schwartz
15007:LF-space
15002:LB-space
14960:FK-space
14930:Complete
14910:BK-space
14835:Relative
14782:(subset)
14774:(subset)
14701:Seminorm
14684:Bilinear
14427:Category
14239:Algebras
14121:Theorems
14078:Complete
14047:Schwartz
13993:glossary
13785:weighted
13655:Hilbert
13632:Bs space
13502:Examples
13467:Interior
13443:Relative
13421:Zonotope
13400:(subset)
13378:(subset)
13329:Strongly
13308:Lebesgue
13303:Measures
13173:Analysis
13019:Theorems
12970:Spectrum
12895:positive
12878:operator
12816:operator
12806:Bilinear
12771:operator
12754:operator
12734:Operator
12631:Complete
12579:Strictly
12436:(2013).
12269:17499190
12128:13064804
11663:Seminorm
11645:Paranorm
11568:See also
11466:normable
10756:Seminorm
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9738:‖
9704:converge
9667:sequence
9658:topology
9653:-norm).
9541:seminorm
9301:For the
8309:and let
8179:below).
6174:denotes
4468:distance
4371:New York
3222:Octonion
2880:distance
2850:distance
2655:over an
2601:quares.
1746:Examples
1557:Notation
1403:implies
1038:positive
1034:positive
1000:for all
821:seminorm
535:for all
435:for all
198:subfield
173:Given a
122:seminorm
75:commutes
63:function
15107:)
15055:)
14997:K-space
14982:Hilbert
14965:Fréchet
14950:F-space
14925:Brauner
14918:)
14903:)
14885:Asplund
14867:)
14837:)
14757:Bounded
14652:Compact
14637:Bounded
14574: (
14230:Unitary
14210:Nuclear
14195:Compact
14190:Bounded
14185:Adjoint
14159:Min–max
14052:Sobolev
14037:Nuclear
14027:Hilbert
14022:Fréchet
13987: (
13650:Hardy H
13553:c space
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12992:of ODEs
12937:Unitary
12912:Nuclear
12843:Compact
12833:Bounded
12801:Adjoint
12641:Fréchet
12636:F-space
12607: (
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12556:)
12536:Hilbert
12509:Asplund
12304:8588370
12142:Algebra
12120:0920371
11838:page 20
11817:page 28
11420:bounded
11398:. Then
10997:infimum
10995:is the
9124:holds:
6747:F-space
4537:L space
4013:of the
3181:is the
2967:modulus
2961:is the
2941:-sphere
2647:is the
2353:On the
1459:), and
196:over a
113:of the
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15105:Quasi-
15027:Montel
15017:Mackey
14916:Ultra-
14895:Banach
14803:Radial
14767:Convex
14737:Affine
14679:Linear
14647:Closed
14471:(TVSs)
14205:Normal
14042:Orlicz
14032:Hölder
14012:Banach
14001:Spaces
13989:topics
13566:Besov
13406:Radial
13371:Convex
13356:Affine
13325:Weakly
13318:Vector
13190:bundle
12980:radius
12907:Normal
12873:kernel
12838:Closed
12761:Strong
12679:Normed
12669:Mackey
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12496:topics
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7430:
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7218:spaces
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6979:, the
6852:F-norm
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2158:where
107:length
99:2-norm
97:, the
71:origin
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15062:Riesz
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14865:Quasi
14859:Polar
14017:Besov
13641:with
13488:Quasi
13482:Polar
13286:Borel
13237:quasi
12766:polar
12749:polar
12563:Riesz
11903:(PDF)
11784:(3).
11191:that
10859:gauge
10271:then
9811:. If
9809:balls
9329:norms
8633:does
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8175:(see
5207:. On
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2852:, or
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1357:" is
1195:with
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659:then
268:is a
85:in a
73:: it
61:is a
14696:Norm
14620:form
14608:Maps
14365:(or
14083:Dual
13639:C(K)
13274:weak
12811:form
12744:Weak
12717:Dual
12684:norm
12646:tame
12519:list
12452:OCLC
12442:ISBN
12422:OCLC
12412:ISBN
12392:OCLC
12382:ISBN
12358:OCLC
12348:ISBN
12331:OCLC
12321:ISBN
12300:OCLC
12290:ISBN
12265:OCLC
12255:ISBN
12189:ISBN
12146:ISBN
12124:OCLC
12106:ISBN
11975:ISBN
11914:2020
11821:ISBN
11794:ISSN
11704:ISBN
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