771:
355:
1171:
Linear operators also play a great role in the infinite-dimensional case. The concepts of rank and determinant cannot be extended to infinite-dimensional matrices. This is why very different techniques are employed when studying linear operators (and operators in general) in the infinite-dimensional
1964:
The
Fourier transform is useful in applied mathematics, particularly physics and signal processing. It is another integral operator; it is useful mainly because it converts a function on one (temporal) domain to a function on another (frequency) domain, in a way effectively
2346:
1474:
1183:
of real numbers, or more generally sequences of vectors in any vector space, themselves form an infinite-dimensional vector space. The most important cases are sequences of real or complex numbers, and these spaces, together with linear subspaces, are known as
390:
This means that a linear operator preserves vector space operations, in the sense that it does not matter whether you apply the linear operator before or after the operations of addition and scalar multiplication. In more technical words, linear operators are
2136:
2510:
1546:
648:
271:
841:
1352:
2228:
1375:
1980:
1040:
482:
1652:
536:
1759:) assigns a vector at every point in a scalar field that points in the direction of greatest rate of change of that field and whose norm measures the absolute value of that greatest rate of change.
2389:
587:
952:
2216:
643:
260:
1095:
1486:
899:
1822:
148:
119:
1790:
879:
1689:
1261:
1289:
984:
1756:
766:{\displaystyle \ \operatorname {A} \mathbf {x} =x^{i}\operatorname {A} \mathbf {u} _{i}=x^{i}\left(\operatorname {A} \mathbf {u} _{i}\right)^{j}\mathbf {v} _{j}~.}
350:{\displaystyle \operatorname {A} \left(\alpha \mathbf {x} +\beta \mathbf {y} \right)=\alpha \operatorname {A} \mathbf {x} +\beta \operatorname {A} \mathbf {y} \ }
1147:
1127:
1004:
607:
421:
69:
is a set of functions or other structured objects. Also, the domain of an operator is often difficult to characterize explicitly (for example in the case of an
776:
1829:
As an extension of vector calculus operators to physics, engineering and tensor spaces, grad, div and curl operators also are often associated with
57:
to produce elements of another space (possibly and sometimes required to be the same space). There is no general definition of an
1009:
434:
2341:{\displaystyle f(t)={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{+\infty }{g(\omega )\ e^{i\ \omega \ t}\ \mathrm {d} \ \omega }}
1613:
491:
1942:; every standard deviation is a norm (square root of the quadratic norm); the corresponding cosine to this dot product is the
1934:, which are used to name both number statistics and the operators which produce them. Indeed, every covariance is basically a
2722:
1364:. Bounded operators form a vector space. On this vector space we can introduce a norm that is compatible with the norms of
1825:) is a vector operator that measures a vector field's curling (winding around, rotating around) trend about a given point.
1973:, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of
545:
1469:{\displaystyle \|\operatorname {A} \|=\inf\{\ c:\|\operatorname {A} \mathbf {x} \|_{V}\leq c\ \|\mathbf {x} \|_{U}\}.}
2747:
2675:
2131:{\displaystyle f(t)={\frac {\ a_{0}\ }{2}}+\sum _{n=1}^{\infty }\ a_{n}\cos(\omega \ n\ t)+b_{n}\sin(\omega \ n\ t)}
904:
2191:
1176:(so called because various classes of functions form interesting examples of infinite-dimensional vector spaces).
1943:
185:
2778:
2566:. ... We should also mention that some authors use the term linear operator for any linear transformation from
1204:
616:
233:
2505:{\displaystyle F(s)=\operatorname {\mathcal {L}} \{f\}(s)=\int _{0}^{\infty }e^{-s\ t}\ f(t)\ \mathrm {d} \ t}
1152:
The important concepts directly related to operators between finite-dimensional vector spaces are the ones of
155:
1541:{\textstyle \|\operatorname {A} \operatorname {B} \|\leq \|\operatorname {A} \|\cdot \|\operatorname {B} \|}
2773:
2620:
In this expression, the raised dot merely represents multiplication in whatever scalar field is used with
2366:
is another integral operator and is involved in simplifying the process of solving differential equations.
1793:) is a vector operator that measures a vector field's divergence from or convergence towards a given point.
1203:
in respect to the standard operator norm. The theory of Banach algebras develops a very general concept of
73:), and may be extended so as to act on related objects (an operator that acts on functions may act also on
1045:
884:
1803:
124:
95:
1885:
form a vector space under operator addition; since, for example, both the identity and −identity are
1770:
20:
846:
92:. Linear operators refer to linear maps whose domain and range are the same space, for example from
1946:; expected value is basically an integral operator (used to measure weighted shapes in the space).
1905:
1662:
1244:
1904:. Operators in the orthogonal group that also preserve the orientation of vector tuples form the
1189:
2637:
957:
177:
159:
50:
27:
1870:
For example, bijective operators preserving the structure of a vector space are precisely the
1741:
1608:
1590:
395:
between vector spaces. In the finite-dimensional case linear operators can be represented by
163:
74:
1969:. No information is lost, as there is an inverse transform operator. In the simple case of
1878:
1842:
396:
263:
181:
66:
8:
1966:
1871:
1864:
1600:
1173:
610:
219:
151:
54:
31:
1917:
1796:
1720:
1264:
1132:
1112:
989:
592:
406:
78:
2743:
2718:
2671:
2647:
2357:
1970:
1959:
1656:
1594:
1573:
226:
167:
70:
46:
2768:
2642:
1901:
1893:
1216:
1161:
836:{\displaystyle a_{i}^{j}\equiv \left(\operatorname {A} \mathbf {u} _{i}\right)^{j}}
1347:{\displaystyle \|\operatorname {A} \mathbf {x} \|_{V}\leq c\ \|\mathbf {x} \|_{U}}
1874:
1830:
1726:
1700:
1569:
1561:
1172:
case. The study of linear operators in the infinite-dimensional case is known as
197:
2178:
1955:
1939:
1923:
1897:
1557:
1553:
1224:
1200:
1185:
2762:
1694:
1238:
1220:
1859:
are sometimes studied. Operators that map such vector spaces to themselves
1856:
1708:
1704:
1196:
215:
89:
1938:: Every variance is a dot product of a vector with itself, and thus is a
1935:
1157:
1153:
38:
1931:
1886:
1763:
1716:
1565:
1165:
85:
1974:
1860:
1106:
1767:
162:
are linear operators; operators that are built from them are called
19:
This article is about operators in mathematics. For other uses, see
2666:
Rudin, Walter (1976). "Chapter 9: Functions of several variables".
2219:
1927:
1852:
1846:
1734:
1712:
1604:
1180:
392:
77:
whose solutions are functions that satisfy the equation). (see
2177:
is in fact an element of an infinite-dimensional vector space
1800:
2558: . A linear operator on a real vector space is called a
1900:, and those that fix the origin form a subgroup known as the
2562:
and a linear operator on a complex vector space is called a
2713:
1922:
Operators are also involved in probability theory, such as
1035:{\displaystyle \operatorname {A} \mathbf {x} =\mathbf {y} }
477:{\displaystyle \ \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}}
1695:
Fundamental analysis operators on scalar and vector fields
1572:
with some additional structure, play an important role in
1738:
1647:{\displaystyle {\frac {\ \mathrm {d} \ }{\mathrm {d} t}}}
531:{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{m}}
1949:
1863:
are very useful in these studies, they naturally form
1489:
1207:
that elegantly generalizes the theory of eigenspaces.
2392:
2231:
2194:
1983:
1806:
1773:
1744:
1665:
1616:
1378:
1292:
1247:
1135:
1115:
1048:
1012:
992:
960:
907:
887:
849:
779:
651:
619:
595:
548:
494:
437:
409:
274:
236:
127:
98:
180:. This is related with the meaning of "operator" in
150:. Such operators often preserve properties, such as
2708:
2706:
2659:
2504:
2340:
2210:
2130:
1816:
1784:
1750:
1683:
1646:
1540:
1468:
1346:
1255:
1141:
1121:
1089:
1034:
998:
978:
946:
893:
873:
835:
765:
637:
601:
582:{\displaystyle \mathbf {x} =x^{i}\mathbf {u} _{i}}
581:
530:
476:
415:
349:
254:
202:The most common kind of operators encountered are
142:
113:
1810:
2760:
2703:
2185:, and thus Fourier series is a linear operator.
1394:
947:{\displaystyle \{\mathbf {u} _{i}\}_{i=1}^{n}}
26:Not to be confused with the symbol denoting a
2424:
2418:
2211:{\displaystyle \mathbb {R} \to \mathbb {C} }
1535:
1526:
1520:
1511:
1505:
1490:
1460:
1451:
1442:
1424:
1409:
1397:
1388:
1379:
1335:
1326:
1308:
1293:
924:
908:
2670:(3rd ed.). McGraw-Hill. p. 207.
1607:is the study of two linear operators: the
176:is also used for denoting the symbol of a
2731:
2204:
2196:
1249:
1188:. Operators on these spaces are known as
638:{\displaystyle \operatorname {A} :U\to V}
427:be finite-dimensional vector spaces over
255:{\displaystyle \operatorname {A} :U\to V}
130:
101:
61:, but the term is often used in place of
1109:correspondence to linear operators from
2582:The following terms are also employed:
1889:(bijective), but their sum, 0, is not.
2761:
2717:(3rd ed.). Springer. p. 59.
1584:
2737:
2712:
2665:
2543:. The set of all linear operators on
1911:
881:, is the matrix form of the operator
2351:
1950:Fourier series and Fourier transform
1210:
1090:{\displaystyle a_{i}^{j}x^{i}=y^{j}}
2668:Principles of Mathematical Analysis
2527:: (1) A linear transformation from
2188:When dealing with general function
1237:be two vector spaces over the same
191:
170:or integro-differential operators.
13:
2492:
2449:
2410:
2327:
2280:
2272:
2043:
1807:
1775:
1745:
1634:
1624:
1529:
1514:
1499:
1493:
1412:
1382:
1296:
1013:
894:{\displaystyle \operatorname {A} }
888:
804:
716:
679:
655:
620:
333:
316:
275:
237:
16:Function acting on function spaces
14:
2790:
1881:under composition. However, they
986:does not depend on the choice of
1817:{\displaystyle \nabla \!\times }
1446:
1419:
1330:
1303:
1195:Bounded linear operators over a
1028:
1020:
913:
812:
747:
724:
687:
662:
569:
550:
518:
497:
464:
443:
340:
323:
301:
290:
143:{\displaystyle \mathbb {R} ^{n}}
114:{\displaystyle \mathbb {R} ^{n}}
2596:for bijective linear operator.
1944:Pearson correlation coefficient
1785:{\displaystyle {\nabla \cdot }}
1560:. It is possible to generalize
1556:with this property is called a
1480:to itself it can be shown that
186:Operator (computer programming)
2614:
2521:
2485:
2479:
2433:
2427:
2402:
2396:
2295:
2289:
2241:
2235:
2200:
2125:
2107:
2085:
2067:
1993:
1987:
1267:. Then a linear operator from
1263:), and they are equipped with
874:{\displaystyle a_{i}^{j}\in K}
629:
246:
1:
2742:. New York, NY: W.W. Norton.
2740:Div, Grad, Curl, and All That
2653:
1908:, or the group of rotations.
1684:{\displaystyle \int _{0}^{t}}
84:The most basic operators are
2514:
2218:, the transform takes on an
1833:as well as vector calculus.
1256:{\displaystyle \mathbb {R} }
7:
2631:
1855:, additional structures on
1836:
1725:Three operators are key to
1579:
53:that acts on elements of a
10:
2795:
2682:Linear transformations of
2355:
1953:
1915:
1840:
1698:
1599:From the point of view of
1588:
1476:In case of operators from
1214:
645:be a linear operator. Then
589:be an arbitrary vector in
399:in the following way. Let
195:
25:
18:
1896:on such a space form the
1892:Operators preserving the
1766:), (with operator symbol
1737:), (with operator symbol
979:{\displaystyle a_{i}^{j}}
21:Operator (disambiguation)
1906:special orthogonal group
1799:, (with operator symbol
1190:sequence transformations
431:. Let us select a basis
2715:Advanced Linear Algebra
2589:for linear operator ...
1751:{\displaystyle \nabla }
2506:
2342:
2212:
2132:
2047:
1818:
1786:
1752:
1685:
1648:
1542:
1470:
1348:
1257:
1143:
1123:
1097:. Thus in fixed bases
1091:
1036:
1000:
980:
948:
895:
875:
837:
767:
639:
603:
583:
532:
478:
417:
351:
256:
178:mathematical operation
164:differential operators
160:indefinite integration
144:
115:
75:differential equations
28:mathematical operation
2779:Mathematical notation
2507:
2343:
2213:
2133:
2027:
1819:
1787:
1753:
1686:
1649:
1609:differential operator
1591:Differential operator
1543:
1471:
1349:
1258:
1144:
1124:
1092:
1037:
1001:
981:
949:
896:
876:
838:
768:
640:
604:
584:
533:
479:
418:
352:
257:
145:
116:
2738:Schey, H.M. (2005).
2390:
2229:
2192:
1981:
1879:general linear group
1843:General linear group
1804:
1771:
1742:
1663:
1614:
1487:
1376:
1290:
1245:
1133:
1113:
1046:
1010:
990:
958:
905:
885:
847:
777:
649:
617:
593:
546:
492:
435:
407:
272:
234:
182:computer programming
125:
96:
81:for other examples)
2774:Functional analysis
2453:
2386:, it is defined by:
2284:
1680:
1601:functional analysis
1585:Analysis (calculus)
1174:functional analysis
1063:
975:
943:
901:in the fixed basis
864:
794:
611:Einstein convention
32:mathematical symbol
2502:
2439:
2338:
2264:
2208:
2128:
1971:periodic functions
1918:Probability theory
1912:Probability theory
1814:
1782:
1748:
1681:
1666:
1644:
1564:to such algebras.
1538:
1466:
1344:
1253:
1139:
1119:
1087:
1049:
1032:
996:
976:
961:
944:
923:
891:
871:
850:
833:
780:
763:
635:
599:
579:
528:
474:
413:
347:
252:
168:integral operators
140:
111:
79:Operator (physics)
2724:978-0-387-72828-5
2690:are often called
2648:List of operators
2498:
2490:
2475:
2467:
2364:Laplace transform
2358:Laplace transform
2352:Laplace transform
2333:
2325:
2317:
2311:
2300:
2262:
2260:
2121:
2115:
2081:
2075:
2050:
2022:
2017:
2004:
1977:and cosine waves:
1960:Fourier transform
1657:Volterra operator
1642:
1630:
1622:
1595:Integral operator
1574:quantum mechanics
1441:
1402:
1325:
1211:Bounded operators
1142:{\displaystyle V}
1122:{\displaystyle U}
999:{\displaystyle x}
759:
654:
602:{\displaystyle U}
440:
416:{\displaystyle U}
346:
71:integral operator
2786:
2754:
2753:
2735:
2729:
2728:
2710:
2701:
2700:
2697:
2692:linear operators
2689:
2685:
2663:
2643:Operator algebra
2625:
2623:
2618:
2612:
2573:
2569:
2564:complex operator
2557:
2546:
2542:
2534:
2530:
2525:
2511:
2509:
2508:
2503:
2496:
2495:
2488:
2473:
2472:
2471:
2465:
2452:
2447:
2414:
2413:
2385:
2384:
2347:
2345:
2344:
2339:
2337:
2331:
2330:
2323:
2322:
2321:
2315:
2309:
2298:
2283:
2275:
2263:
2261:
2253:
2248:
2217:
2215:
2214:
2209:
2207:
2199:
2183:
2176:
2137:
2135:
2134:
2129:
2119:
2113:
2100:
2099:
2079:
2073:
2060:
2059:
2048:
2046:
2041:
2023:
2018:
2015:
2014:
2013:
2002:
2000:
1902:orthogonal group
1894:Euclidean metric
1877:. They form the
1875:linear operators
1867:by composition.
1823:
1821:
1820:
1815:
1791:
1789:
1788:
1783:
1781:
1757:
1755:
1754:
1749:
1690:
1688:
1687:
1682:
1679:
1674:
1653:
1651:
1650:
1645:
1643:
1641:
1637:
1631:
1628:
1627:
1620:
1618:
1547:
1545:
1544:
1539:
1479:
1475:
1473:
1472:
1467:
1459:
1458:
1449:
1439:
1432:
1431:
1422:
1400:
1371:
1367:
1363:
1358:
1353:
1351:
1350:
1345:
1343:
1342:
1333:
1323:
1316:
1315:
1306:
1285:
1279:if there exists
1274:
1270:
1262:
1260:
1259:
1254:
1252:
1236:
1232:
1217:Bounded operator
1162:inverse operator
1148:
1146:
1145:
1140:
1128:
1126:
1125:
1120:
1105:matrices are in
1104:
1100:
1096:
1094:
1093:
1088:
1086:
1085:
1073:
1072:
1062:
1057:
1041:
1039:
1038:
1033:
1031:
1023:
1005:
1003:
1002:
997:
985:
983:
982:
977:
974:
969:
953:
951:
950:
945:
942:
937:
922:
921:
916:
900:
898:
897:
892:
880:
878:
877:
872:
863:
858:
842:
840:
839:
834:
832:
831:
826:
822:
821:
820:
815:
793:
788:
772:
770:
769:
764:
757:
756:
755:
750:
744:
743:
738:
734:
733:
732:
727:
709:
708:
696:
695:
690:
678:
677:
665:
652:
644:
642:
641:
636:
608:
606:
605:
600:
588:
586:
585:
580:
578:
577:
572:
566:
565:
553:
541:
537:
535:
534:
529:
527:
526:
521:
506:
505:
500:
487:
483:
481:
480:
475:
473:
472:
467:
452:
451:
446:
438:
430:
426:
422:
420:
419:
414:
403:be a field, and
402:
386:
382:
372:
368:
362:
356:
354:
353:
348:
344:
343:
326:
309:
305:
304:
293:
261:
259:
258:
253:
224:
213:
209:
204:linear operators
192:Linear operators
149:
147:
146:
141:
139:
138:
133:
120:
118:
117:
112:
110:
109:
104:
2794:
2793:
2789:
2788:
2787:
2785:
2784:
2783:
2759:
2758:
2757:
2750:
2736:
2732:
2725:
2711:
2704:
2695:
2687:
2683:
2678:
2664:
2660:
2656:
2634:
2629:
2628:
2621:
2619:
2615:
2571:
2567:
2548:
2544:
2540:
2537:linear operator
2532:
2528:
2526:
2522:
2517:
2491:
2458:
2454:
2448:
2443:
2409:
2408:
2391:
2388:
2387:
2371:
2370:
2360:
2354:
2326:
2305:
2301:
2285:
2276:
2268:
2252:
2247:
2230:
2227:
2226:
2203:
2195:
2193:
2190:
2189:
2179:
2174:
2167:
2160:
2153:
2146:
2139:
2095:
2091:
2055:
2051:
2042:
2031:
2009:
2005:
2001:
1999:
1982:
1979:
1978:
1962:
1954:Main articles:
1952:
1920:
1914:
1849:
1841:Main articles:
1839:
1831:tensor calculus
1805:
1802:
1801:
1774:
1772:
1769:
1768:
1743:
1740:
1739:
1727:vector calculus
1723:
1701:Vector calculus
1699:Main articles:
1697:
1675:
1670:
1664:
1661:
1660:
1633:
1632:
1623:
1619:
1617:
1615:
1612:
1611:
1597:
1589:Main articles:
1587:
1582:
1570:Banach algebras
1562:spectral theory
1488:
1485:
1484:
1477:
1454:
1450:
1445:
1427:
1423:
1418:
1377:
1374:
1373:
1369:
1365:
1361:
1356:
1338:
1334:
1329:
1311:
1307:
1302:
1291:
1288:
1287:
1280:
1272:
1268:
1248:
1246:
1243:
1242:
1234:
1230:
1227:
1215:Main articles:
1213:
1186:sequence spaces
1134:
1131:
1130:
1114:
1111:
1110:
1102:
1098:
1081:
1077:
1068:
1064:
1058:
1053:
1047:
1044:
1043:
1027:
1019:
1011:
1008:
1007:
991:
988:
987:
970:
965:
959:
956:
955:
938:
927:
917:
912:
911:
906:
903:
902:
886:
883:
882:
859:
854:
848:
845:
844:
827:
816:
811:
810:
803:
799:
798:
789:
784:
778:
775:
774:
751:
746:
745:
739:
728:
723:
722:
715:
711:
710:
704:
700:
691:
686:
685:
673:
669:
661:
650:
647:
646:
618:
615:
614:
594:
591:
590:
573:
568:
567:
561:
557:
549:
547:
544:
543:
539:
522:
517:
516:
501:
496:
495:
493:
490:
489:
485:
468:
463:
462:
447:
442:
441:
436:
433:
432:
428:
424:
408:
405:
404:
400:
384:
374:
370:
364:
358:
339:
322:
300:
289:
285:
281:
273:
270:
269:
235:
232:
231:
222:
211:
207:
200:
198:Linear operator
194:
156:differentiation
154:. For example,
134:
129:
128:
126:
123:
122:
105:
100:
99:
97:
94:
93:
88:, which act on
45:is generally a
35:
24:
17:
12:
11:
5:
2792:
2782:
2781:
2776:
2771:
2756:
2755:
2748:
2730:
2723:
2702:
2676:
2657:
2655:
2652:
2651:
2650:
2645:
2640:
2633:
2630:
2627:
2626:
2613:
2611:
2610:
2609:
2608:
2607:
2606:
2605:
2604:
2603:— Roman (2008)
2590:
2578:
2577:
2519:
2518:
2516:
2513:
2501:
2494:
2487:
2484:
2481:
2478:
2470:
2464:
2461:
2457:
2451:
2446:
2442:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2412:
2407:
2404:
2401:
2398:
2395:
2356:Main article:
2353:
2350:
2349:
2348:
2336:
2329:
2320:
2314:
2308:
2304:
2297:
2294:
2291:
2288:
2282:
2279:
2274:
2271:
2267:
2259:
2256:
2251:
2246:
2243:
2240:
2237:
2234:
2206:
2202:
2198:
2172:
2165:
2158:
2151:
2144:
2127:
2124:
2118:
2112:
2109:
2106:
2103:
2098:
2094:
2090:
2087:
2084:
2078:
2072:
2069:
2066:
2063:
2058:
2054:
2045:
2040:
2037:
2034:
2030:
2026:
2021:
2012:
2008:
1998:
1995:
1992:
1989:
1986:
1956:Fourier series
1951:
1948:
1940:quadratic norm
1916:Main article:
1913:
1910:
1898:isometry group
1838:
1835:
1827:
1826:
1813:
1809:
1794:
1780:
1777:
1760:
1747:
1696:
1693:
1678:
1673:
1669:
1640:
1636:
1626:
1586:
1583:
1581:
1578:
1558:Banach algebra
1554:normed algebra
1550:
1549:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1465:
1462:
1457:
1453:
1448:
1444:
1438:
1435:
1430:
1426:
1421:
1417:
1414:
1411:
1408:
1405:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1341:
1337:
1332:
1328:
1322:
1319:
1314:
1310:
1305:
1301:
1298:
1295:
1251:
1241:(for example;
1225:Banach algebra
1212:
1209:
1201:Banach algebra
1138:
1118:
1084:
1080:
1076:
1071:
1067:
1061:
1056:
1052:
1030:
1026:
1022:
1018:
1015:
995:
973:
968:
964:
941:
936:
933:
930:
926:
920:
915:
910:
890:
870:
867:
862:
857:
853:
830:
825:
819:
814:
809:
806:
802:
797:
792:
787:
783:
762:
754:
749:
742:
737:
731:
726:
721:
718:
714:
707:
703:
699:
694:
689:
684:
681:
676:
672:
668:
664:
660:
657:
634:
631:
628:
625:
622:
598:
576:
571:
564:
560:
556:
552:
525:
520:
515:
512:
509:
504:
499:
471:
466:
461:
458:
455:
450:
445:
412:
373:, and for all
342:
338:
335:
332:
329:
325:
321:
318:
315:
312:
308:
303:
299:
296:
292:
288:
284:
280:
277:
251:
248:
245:
242:
239:
196:Main article:
193:
190:
137:
132:
108:
103:
15:
9:
6:
4:
3:
2:
2791:
2780:
2777:
2775:
2772:
2770:
2767:
2766:
2764:
2751:
2749:0-393-92516-1
2745:
2741:
2734:
2726:
2720:
2716:
2709:
2707:
2699:
2693:
2679:
2677:0-07-054235-X
2673:
2669:
2662:
2658:
2649:
2646:
2644:
2641:
2639:
2636:
2635:
2617:
2602:
2601:
2600:
2599:
2598:
2597:
2595:
2591:
2588:
2584:
2583:
2581:
2576:
2575:
2565:
2561:
2560:real operator
2555:
2551:
2538:
2524:
2520:
2512:
2499:
2482:
2476:
2468:
2462:
2459:
2455:
2444:
2440:
2436:
2430:
2421:
2415:
2405:
2399:
2393:
2382:
2378:
2374:
2367:
2365:
2359:
2334:
2318:
2312:
2306:
2302:
2292:
2286:
2277:
2269:
2265:
2257:
2254:
2249:
2244:
2238:
2232:
2225:
2224:
2223:
2221:
2186:
2184:
2182:
2171:
2164:
2157:
2150:
2143:
2122:
2116:
2110:
2104:
2101:
2096:
2092:
2088:
2082:
2076:
2070:
2064:
2061:
2056:
2052:
2038:
2035:
2032:
2028:
2024:
2019:
2010:
2006:
1996:
1990:
1984:
1976:
1972:
1968:
1961:
1957:
1947:
1945:
1941:
1937:
1933:
1929:
1925:
1919:
1909:
1907:
1903:
1899:
1895:
1890:
1888:
1884:
1880:
1876:
1873:
1868:
1866:
1862:
1858:
1857:vector spaces
1854:
1848:
1844:
1834:
1832:
1824:
1811:
1798:
1795:
1792:
1778:
1765:
1761:
1758:
1736:
1732:
1731:
1730:
1728:
1722:
1718:
1714:
1710:
1706:
1702:
1692:
1676:
1671:
1667:
1659:
1658:
1638:
1610:
1606:
1602:
1596:
1592:
1577:
1575:
1571:
1567:
1563:
1559:
1555:
1532:
1523:
1517:
1508:
1502:
1496:
1483:
1482:
1481:
1463:
1455:
1436:
1433:
1428:
1415:
1406:
1403:
1391:
1385:
1359:
1339:
1320:
1317:
1312:
1299:
1283:
1278:
1266:
1240:
1239:ordered field
1226:
1222:
1221:Operator norm
1218:
1208:
1206:
1202:
1198:
1193:
1191:
1187:
1182:
1179:The space of
1177:
1175:
1169:
1167:
1163:
1159:
1155:
1150:
1136:
1116:
1108:
1082:
1078:
1074:
1069:
1065:
1059:
1054:
1050:
1024:
1016:
993:
971:
966:
962:
954:. The tensor
939:
934:
931:
928:
918:
868:
865:
860:
855:
851:
828:
823:
817:
807:
800:
795:
790:
785:
781:
760:
752:
740:
735:
729:
719:
712:
705:
701:
697:
692:
682:
674:
670:
666:
658:
632:
626:
623:
612:
596:
574:
562:
558:
554:
523:
513:
510:
507:
502:
469:
459:
456:
453:
448:
410:
398:
394:
388:
381:
377:
367:
361:
336:
330:
327:
319:
313:
310:
306:
297:
294:
286:
282:
278:
267:
266:
249:
243:
240:
230:
229:
221:
217:
216:vector spaces
205:
199:
189:
187:
183:
179:
175:
171:
169:
165:
161:
157:
153:
135:
106:
91:
90:vector spaces
87:
82:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
33:
29:
22:
2739:
2733:
2714:
2691:
2681:
2667:
2661:
2616:
2594:automorphism
2593:
2587:endomorphism
2586:
2579:
2563:
2559:
2553:
2549:
2536:
2535:is called a
2523:
2380:
2376:
2372:
2368:
2363:
2361:
2187:
2180:
2169:
2162:
2155:
2148:
2141:
1963:
1921:
1891:
1882:
1869:
1850:
1828:
1724:
1709:Scalar field
1705:Vector field
1655:
1598:
1568:, which are
1551:
1355:
1281:
1276:
1228:
1197:Banach space
1194:
1178:
1170:
1151:
389:
379:
375:
365:
359:
264:
227:
203:
201:
173:
172:
83:
62:
58:
42:
36:
2580:Definition:
2547:is denoted
1936:dot product
1924:expectation
1861:bijectively
1566:C*-algebras
1552:Any unital
1158:determinant
843:, with all
542:. Then let
86:linear maps
39:mathematics
2763:Categories
2654:References
2138:The tuple
1975:sine waves
1967:invertible
1932:covariance
1887:invertible
1872:invertible
1764:divergence
1717:Divergence
1654:, and the
1354:for every
1286:such that
1275:is called
1166:eigenspace
609:(assuming
218:over some
152:continuity
2515:Footnotes
2460:−
2450:∞
2441:∫
2416:
2335:ω
2313:ω
2293:ω
2281:∞
2273:∞
2270:−
2266:∫
2258:π
2201:→
2111:ω
2105:
2071:ω
2065:
2044:∞
2029:∑
1812:×
1808:∇
1779:⋅
1776:∇
1746:∇
1668:∫
1536:‖
1533:
1527:‖
1524:⋅
1521:‖
1518:
1512:‖
1509:≤
1506:‖
1503:
1497:
1491:‖
1452:‖
1443:‖
1434:≤
1425:‖
1416:
1410:‖
1389:‖
1386:
1380:‖
1336:‖
1327:‖
1318:≤
1309:‖
1300:
1294:‖
1181:sequences
1107:bijective
1017:
866:∈
808:
796:≡
720:
683:
659:
630:→
511:…
457:…
393:morphisms
337:
331:β
320:
314:α
298:β
287:α
279:
247:→
65:when the
2638:Function
2632:See also
2624: .
2220:integral
1928:variance
1853:geometry
1847:Isometry
1837:Geometry
1735:gradient
1713:Gradient
1605:calculus
1580:Examples
397:matrices
357:for all
174:Operator
63:function
59:operator
51:function
43:operator
2769:Algebra
2698: .
2175:, ... )
1277:bounded
1205:spectra
1199:form a
613:), and
228:mapping
47:mapping
2746:
2721:
2674:
2574:. ...
2497:
2489:
2474:
2466:
2369:Given
2332:
2324:
2316:
2310:
2299:
2222:form:
2120:
2114:
2080:
2074:
2049:
2016:
2003:
1930:, and
1883:do not
1865:groups
1733:Grad (
1719:, and
1629:
1621:
1440:
1401:
1324:
1284:> 0
1223:, and
1164:, and
1006:, and
758:
653:
439:
345:
265:linear
206:. Let
67:domain
2686:into
1762:Div (
1265:norms
773:Then
220:field
184:(see
55:space
41:, an
2744:ISBN
2719:ISBN
2672:ISBN
2592:(6)
2585:(2)
2362:The
1958:and
1845:and
1797:Curl
1721:Curl
1593:and
1368:and
1233:and
1229:Let
1154:rank
1101:-by-
488:and
423:and
363:and
225:. A
210:and
158:and
2694:on
2570:to
2539:on
2531:to
2102:sin
2062:cos
1851:In
1691:.
1395:inf
1360:in
1271:to
1149:.
1129:to
1042:if
538:in
484:in
383:in
369:in
268:if
262:is
214:be
188:).
121:to
49:or
37:In
30:or
2765::
2705:^
2680:.
2375:=
2168:,
2161:,
2154:,
2147:,
2140:(
1926:,
1729::
1715:,
1711:,
1707:,
1703:,
1603:,
1576:.
1372::
1219:,
1192:.
1168:.
1160:,
1156:,
387:.
378:,
166:,
2752:.
2727:.
2696:X
2688:X
2684:X
2622:V
2572:W
2568:V
2556:)
2554:V
2552:(
2550:ℒ
2545:V
2541:V
2533:V
2529:V
2500:t
2493:d
2486:)
2483:t
2480:(
2477:f
2469:t
2463:s
2456:e
2445:0
2437:=
2434:)
2431:s
2428:(
2425:}
2422:f
2419:{
2411:L
2406:=
2403:)
2400:s
2397:(
2394:F
2383:)
2381:s
2379:(
2377:f
2373:f
2328:d
2319:t
2307:i
2303:e
2296:)
2290:(
2287:g
2278:+
2255:2
2250:1
2245:=
2242:)
2239:t
2236:(
2233:f
2205:C
2197:R
2181:ℓ
2173:2
2170:b
2166:2
2163:a
2159:1
2156:b
2152:1
2149:a
2145:0
2142:a
2126:)
2123:t
2117:n
2108:(
2097:n
2093:b
2089:+
2086:)
2083:t
2077:n
2068:(
2057:n
2053:a
2039:1
2036:=
2033:n
2025:+
2020:2
2011:0
2007:a
1997:=
1994:)
1991:t
1988:(
1985:f
1677:t
1672:0
1639:t
1635:d
1625:d
1548:.
1530:B
1515:A
1500:B
1494:A
1478:U
1464:.
1461:}
1456:U
1447:x
1437:c
1429:V
1420:x
1413:A
1407::
1404:c
1398:{
1392:=
1383:A
1370:V
1366:U
1362:U
1357:x
1340:U
1331:x
1321:c
1313:V
1304:x
1297:A
1282:c
1273:V
1269:U
1250:R
1235:V
1231:U
1137:V
1117:U
1103:m
1099:n
1083:j
1079:y
1075:=
1070:i
1066:x
1060:j
1055:i
1051:a
1029:y
1025:=
1021:x
1014:A
994:x
972:j
967:i
963:a
940:n
935:1
932:=
929:i
925:}
919:i
914:u
909:{
889:A
869:K
861:j
856:i
852:a
829:j
824:)
818:i
813:u
805:A
801:(
791:j
786:i
782:a
761:.
753:j
748:v
741:j
736:)
730:i
725:u
717:A
713:(
706:i
702:x
698:=
693:i
688:u
680:A
675:i
671:x
667:=
663:x
656:A
633:V
627:U
624::
621:A
597:U
575:i
570:u
563:i
559:x
555:=
551:x
540:V
524:m
519:v
514:,
508:,
503:1
498:v
486:U
470:n
465:u
460:,
454:,
449:1
444:u
429:K
425:V
411:U
401:K
385:K
380:β
376:α
371:U
366:y
360:x
341:y
334:A
328:+
324:x
317:A
311:=
307:)
302:y
295:+
291:x
283:(
276:A
250:V
244:U
241::
238:A
223:K
212:V
208:U
136:n
131:R
107:n
102:R
34:.
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.