782:
366:
1182:
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
1975:
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
2357:
1485:
1194:
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
401:
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
2147:
2521:
1557:
659:
282:
852:
1363:
2239:
1386:
1991:
1051:
493:
1663:
547:
1770:) 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.
2400:
598:
963:
2227:
654:
271:
1106:
1497:
910:
1833:
159:
130:
1801:
890:
1700:
1272:
1300:
995:
1767:
777:{\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}~.}
361:{\displaystyle \operatorname {A} \left(\alpha \mathbf {x} +\beta \mathbf {y} \right)=\alpha \operatorname {A} \mathbf {x} +\beta \operatorname {A} \mathbf {y} \ }
1158:
1138:
1015:
618:
432:
80:
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
787:
1840:
As an extension of vector calculus operators to physics, engineering and tensor spaces, grad, div and curl operators also are often associated with
2733:
2758:
2686:
68:
to produce elements of another space (possibly and sometimes required to be the same space). There is no general definition of an
1020:
445:
2352:{\displaystyle f(t)={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{+\infty }{g(\omega )\ e^{i\ \omega \ t}\ \mathrm {d} \ \omega }}
1954:
1624:
502:
196:
1953:; every standard deviation is a norm (square root of the quadratic norm); the corresponding cosine to this dot product is the
1945:, which are used to name both number statistics and the operators which produce them. Indeed, every covariance is basically a
2789:
1215:
1375:. Bounded operators form a vector space. On this vector space we can introduce a norm that is compatible with the norms of
1836:) is a vector operator that measures a vector field's curling (winding around, rotating around) trend about a given point.
166:
1984:, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of
2784:
556:
1480:{\displaystyle \|\operatorname {A} \|=\inf\{\ c:\|\operatorname {A} \mathbf {x} \|_{V}\leq c\ \|\mathbf {x} \|_{U}\}.}
2142:{\displaystyle f(t)={\frac {\ a_{0}\ }{2}}+\sum _{n=1}^{\infty }\ a_{n}\cos(\omega \ n\ t)+b_{n}\sin(\omega \ n\ t)}
915:
2202:
1187:(so called because various classes of functions form interesting examples of infinite-dimensional vector spaces).
31:
2577:. ... We should also mention that some authors use the term linear operator for any linear transformation from
1916:
627:
244:
2516:{\displaystyle F(s)=\operatorname {\mathcal {L}} \{f\}(s)=\int _{0}^{\infty }e^{-s\ t}\ f(t)\ \mathrm {d} \ t}
1163:
The important concepts directly related to operators between finite-dimensional vector spaces are the ones of
1552:{\textstyle \|\operatorname {A} \operatorname {B} \|\leq \|\operatorname {A} \|\cdot \|\operatorname {B} \|}
1200:
2631:
In this expression, the raised dot merely represents multiplication in whatever scalar field is used with
2377:
is another integral operator and is involved in simplifying the process of solving differential equations.
1804:) is a vector operator that measures a vector field's divergence from or convergence towards a given point.
1214:
in respect to the standard operator norm. The theory of Banach algebras develops a very general concept of
84:), and may be extended so as to act on related objects (an operator that acts on functions may act also on
2648:
1056:
188:
170:
61:
38:
1619:
1601:
895:
174:
85:
1814:
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106:
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form a vector space under operator addition; since, for example, both the identity and −identity are
1889:
1853:
1781:
407:
274:
192:
77:
857:
103:. Linear operators refer to linear maps whose domain and range are the same space, for example from
1977:
1957:; expected value is basically an integral operator (used to measure weighted shapes in the space).
1882:
1875:
1673:
1611:
1184:
621:
230:
162:
65:
42:
1255:
1928:
1915:. Operators in the orthogonal group that also preserve the orientation of vector tuples form the
1807:
1731:
1275:
89:
968:
2754:
2729:
2682:
2658:
2368:
1981:
1970:
1881:
For example, bijective operators preserving the structure of a vector space are precisely the
1667:
1605:
1584:
237:
178:
81:
57:
1752:
406:
between vector spaces. In the finite-dimensional case linear operators can be represented by
2779:
2653:
1912:
1904:
1227:
1172:
1980:. No information is lost, as there is an inverse transform operator. In the simple case of
1885:
1841:
1737:
1711:
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208:
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1966:
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1908:
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1123:
1000:
603:
417:
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1249:
1231:
17:
1867:
1719:
1715:
1207:
847:{\displaystyle a_{i}^{j}\equiv \left(\operatorname {A} \mathbf {u} _{i}\right)^{j}}
226:
100:
1358:{\displaystyle \|\operatorname {A} \mathbf {x} \|_{V}\leq c\ \|\mathbf {x} \|_{U}}
1183:
case. The study of linear operators in the infinite-dimensional case is known as
1946:
1168:
1164:
49:
1942:
1897:
1774:
1727:
1576:
1176:
96:
1985:
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1117:
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are sometimes studied. Operators that map such vector spaces to themselves
2230:
1949:: Every variance is a dot product of a vector with itself, and thus is a
1938:
1863:
1857:
1745:
1723:
1615:
1191:
403:
173:
are linear operators; operators that are built from them are called
30:
This article is about operators in mathematics. For other uses, see
2677:
Rudin, Walter (1976). "Chapter 9: Functions of several variables".
88:
whose solutions are functions that satisfy the equation). (see
2188:
is in fact an element of an infinite-dimensional vector space
1811:
1778:
1749:
2569: . A linear operator on a real vector space is called a
1911:, and those that fix the origin form a subgroup known as the
2573:
and a linear operator on a complex vector space is called a
2421:
2724:
1933:
Operators are also involved in probability theory, such as
1046:{\displaystyle \operatorname {A} \mathbf {x} =\mathbf {y} }
488:{\displaystyle \ \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}}
1706:
Fundamental analysis operators on scalar and vector fields
1583:
with some additional structure, play an important role in
1658:{\displaystyle {\frac {\ \mathrm {d} \ }{\mathrm {d} t}}}
542:{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{m}}
1874:
are very useful in these studies, they naturally form
1500:
1218:
that elegantly generalizes the theory of eigenspaces.
2403:
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1994:
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138:
109:
191:. This is related with the meaning of "operator" in
161:. Such operators often preserve properties, such as
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2141:
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957:
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648:
612:
593:{\displaystyle \mathbf {x} =x^{i}\mathbf {u} _{i}}
592:
541:
487:
426:
360:
265:
213:The most common kind of operators encountered are
153:
124:
1821:
2196:, and thus Fourier series is a linear operator.
1405:
958:{\displaystyle \{\mathbf {u} _{i}\}_{i=1}^{n}}
37:Not to be confused with the symbol denoting a
8:
2435:
2429:
2222:{\displaystyle \mathbb {R} \to \mathbb {C} }
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1531:
1522:
1516:
1501:
1471:
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1390:
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1304:
935:
919:
2681:(3rd ed.). McGraw-Hill. p. 207.
1618:is the study of two linear operators: the
187:is also used for denoting the symbol of a
2502:
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2420:
2419:
2402:
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2020:
2010:
1993:
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1438:
1429:
1388:
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1340:
1322:
1313:
1302:
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1259:
1257:
1199:. Operators on these spaces are known as
1145:
1125:
1092:
1079:
1069:
1064:
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1038:
1030:
1022:
1002:
981:
976:
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739:
734:
715:
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684:
672:
661:
649:{\displaystyle \operatorname {A} :U\to V}
629:
605:
584:
579:
572:
560:
558:
533:
528:
512:
507:
504:
479:
474:
458:
453:
447:
438:be finite-dimensional vector spaces over
419:
350:
333:
311:
300:
284:
266:{\displaystyle \operatorname {A} :U\to V}
246:
145:
141:
140:
137:
116:
112:
111:
108:
72:, but the term is often used in place of
1120:correspondence to linear operators from
2669:
2593:The following terms are also employed:
2531:
1900:(bijective), but their sum, 0, is not.
2728:(3rd ed.). Springer. p. 59.
2554:. The set of all linear operators on
892:, is the matrix form of the operator
7:
1961:Fourier series and Fourier transform
1101:{\displaystyle a_{i}^{j}x^{i}=y^{j}}
2679:Principles of Mathematical Analysis
2538:: (1) A linear transformation from
2199:When dealing with general function
1248:be two vector spaces over the same
181:or integro-differential operators.
2503:
2460:
2338:
2291:
2283:
2054:
1818:
1786:
1756:
1645:
1635:
1540:
1525:
1510:
1504:
1423:
1393:
1307:
1024:
905:{\displaystyle \operatorname {A} }
899:
815:
727:
690:
666:
631:
344:
327:
286:
248:
27:Function acting on function spaces
25:
1892:under composition. However, they
997:does not depend on the choice of
1828:{\displaystyle \nabla \!\times }
1457:
1430:
1341:
1314:
1206:Bounded linear operators over a
1039:
1031:
924:
823:
758:
735:
698:
673:
580:
561:
529:
508:
475:
454:
351:
334:
312:
301:
154:{\displaystyle \mathbb {R} ^{n}}
125:{\displaystyle \mathbb {R} ^{n}}
2607:for bijective linear operator.
1955:Pearson correlation coefficient
1796:{\displaystyle {\nabla \cdot }}
1571:. It is possible to generalize
1567:with this property is called a
1491:to itself it can be shown that
197:Operator (computer programming)
2496:
2490:
2444:
2438:
2413:
2407:
2306:
2300:
2252:
2246:
2211:
2136:
2118:
2096:
2078:
2004:
1998:
1278:. Then a linear operator from
1274:), and they are equipped with
885:{\displaystyle a_{i}^{j}\in K}
640:
257:
1:
2753:. New York, NY: W.W. Norton.
2751:Div, Grad, Curl, and All That
1919:, or the group of rotations.
1695:{\displaystyle \int _{0}^{t}}
95:The most basic operators are
2229:, the transform takes on an
1844:as well as vector calculus.
1267:{\displaystyle \mathbb {R} }
1866:, additional structures on
1736:Three operators are key to
64:that acts on elements of a
2806:
2693:Linear transformations of
2366:
1964:
1926:
1851:
1709:
1610:From the point of view of
1599:
1487:In case of operators from
1225:
656:be a linear operator. Then
600:be an arbitrary vector in
410:in the following way. Let
206:
36:
29:
1907:on such a space form the
1903:Operators preserving the
1777:), (with operator symbol
1748:), (with operator symbol
990:{\displaystyle a_{i}^{j}}
32:Operator (disambiguation)
1917:special orthogonal group
1810:, (with operator symbol
1201:sequence transformations
442:. Let us select a basis
2726:Advanced Linear Algebra
2600:for linear operator ...
1762:{\displaystyle \nabla }
2517:
2353:
2223:
2143:
2058:
1829:
1797:
1763:
1696:
1659:
1553:
1481:
1359:
1268:
1154:
1134:
1108:. Thus in fixed bases
1102:
1047:
1011:
991:
959:
906:
886:
848:
778:
650:
614:
594:
543:
489:
428:
362:
267:
189:mathematical operation
175:differential operators
171:indefinite integration
155:
126:
86:differential equations
39:mathematical operation
2790:Mathematical notation
2518:
2354:
2224:
2144:
2038:
1830:
1798:
1764:
1697:
1660:
1620:differential operator
1602:Differential operator
1554:
1482:
1360:
1269:
1155:
1135:
1103:
1048:
1012:
992:
960:
907:
887:
849:
779:
651:
615:
595:
544:
490:
429:
363:
268:
156:
127:
18:Mathematical operator
2749:Schey, H.M. (2005).
2401:
2240:
2203:
1992:
1890:general linear group
1854:General linear group
1815:
1782:
1753:
1674:
1625:
1498:
1387:
1301:
1256:
1144:
1124:
1057:
1021:
1001:
969:
916:
896:
858:
788:
660:
628:
604:
557:
503:
446:
418:
283:
245:
193:computer programming
136:
107:
92:for other examples)
2785:Functional analysis
2464:
2397:, it is defined by:
2295:
1691:
1612:functional analysis
1596:Analysis (calculus)
1185:functional analysis
1074:
986:
954:
912:in the fixed basis
875:
805:
622:Einstein convention
43:mathematical symbol
2513:
2450:
2349:
2275:
2219:
2139:
1982:periodic functions
1929:Probability theory
1923:Probability theory
1825:
1793:
1759:
1692:
1677:
1655:
1575:to such algebras.
1549:
1477:
1355:
1264:
1150:
1130:
1098:
1060:
1043:
1007:
987:
972:
955:
934:
902:
882:
861:
844:
791:
774:
646:
610:
590:
539:
485:
424:
358:
263:
179:integral operators
151:
122:
90:Operator (physics)
2735:978-0-387-72828-5
2701:are often called
2659:List of operators
2509:
2501:
2486:
2478:
2375:Laplace transform
2369:Laplace transform
2363:Laplace transform
2344:
2336:
2328:
2322:
2311:
2273:
2271:
2132:
2126:
2092:
2086:
2061:
2033:
2028:
2015:
1988:and cosine waves:
1971:Fourier transform
1668:Volterra operator
1653:
1641:
1633:
1606:Integral operator
1585:quantum mechanics
1452:
1413:
1336:
1222:Bounded operators
1153:{\displaystyle V}
1133:{\displaystyle U}
1010:{\displaystyle x}
770:
665:
613:{\displaystyle U}
451:
427:{\displaystyle U}
357:
82:integral operator
16:(Redirected from
2797:
2765:
2764:
2746:
2740:
2739:
2721:
2712:
2711:
2708:
2703:linear operators
2700:
2696:
2674:
2654:Operator algebra
2636:
2634:
2629:
2623:
2584:
2580:
2575:complex operator
2568:
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2541:
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2514:
2507:
2506:
2499:
2484:
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2463:
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2228:
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2218:
2210:
2194:
2187:
2148:
2146:
2145:
2140:
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2124:
2111:
2110:
2090:
2084:
2071:
2070:
2059:
2057:
2052:
2034:
2029:
2026:
2025:
2024:
2013:
2011:
1913:orthogonal group
1905:Euclidean metric
1888:. They form the
1886:linear operators
1878:by composition.
1834:
1832:
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1826:
1802:
1800:
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1792:
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1356:
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1344:
1334:
1327:
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1317:
1296:
1290:if there exists
1285:
1281:
1273:
1271:
1270:
1265:
1263:
1247:
1243:
1228:Bounded operator
1173:inverse operator
1159:
1157:
1156:
1151:
1139:
1137:
1136:
1131:
1116:matrices are in
1115:
1111:
1107:
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1084:
1083:
1073:
1068:
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996:
994:
993:
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719:
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689:
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619:
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491:
486:
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463:
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457:
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425:
414:be a field, and
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337:
320:
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215:linear operators
203:Linear operators
160:
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21:
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2645:
2640:
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2626:
2582:
2578:
2559:
2555:
2551:
2548:linear operator
2543:
2539:
2537:
2533:
2528:
2465:
2399:
2398:
2382:
2381:
2371:
2365:
2312:
2238:
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2201:
2200:
2190:
2185:
2178:
2171:
2164:
2157:
2150:
2102:
2062:
2016:
2012:
1990:
1989:
1973:
1965:Main articles:
1963:
1931:
1925:
1860:
1852:Main articles:
1850:
1842:tensor calculus
1813:
1812:
1780:
1779:
1751:
1750:
1738:vector calculus
1734:
1712:Vector calculus
1710:Main articles:
1708:
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1600:Main articles:
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1581:Banach algebras
1573:spectral theory
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1197:sequence spaces
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209:Linear operator
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167:differentiation
165:. For example,
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99:, which act on
56:is generally a
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2614:— Roman (2008)
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1967:Fourier series
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1951:quadratic norm
1927:Main article:
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1909:isometry group
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1569:Banach algebra
1565:normed algebra
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1212:Banach algebra
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1868:vector spaces
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1250:ordered field
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1232:Operator norm
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1190:The space of
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965:. The tensor
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227:vector spaces
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101:vector spaces
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33:
19:
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2744:
2725:
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2678:
2672:
2627:
2605:automorphism
2604:
2598:endomorphism
2597:
2590:
2574:
2570:
2564:
2560:
2547:
2546:is called a
2534:
2391:
2387:
2383:
2379:
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2372:
2198:
2191:
2180:
2173:
2166:
2159:
2152:
1974:
1932:
1902:
1893:
1880:
1861:
1839:
1735:
1720:Scalar field
1716:Vector field
1666:
1609:
1579:, which are
1562:
1366:
1292:
1287:
1239:
1208:Banach space
1205:
1189:
1181:
1162:
400:
390:
386:
376:
370:
275:
238:
214:
212:
184:
183:
94:
73:
69:
53:
47:
2591:Definition:
2558:is denoted
1947:dot product
1935:expectation
1872:bijectively
1577:C*-algebras
1563:Any unital
1169:determinant
854:, with all
553:. Then let
97:linear maps
50:mathematics
2774:Categories
2665:References
2149:The tuple
1986:sine waves
1978:invertible
1943:covariance
1898:invertible
1883:invertible
1775:divergence
1728:Divergence
1665:, and the
1365:for every
1297:such that
1286:is called
1177:eigenspace
620:(assuming
229:over some
163:continuity
2526:Footnotes
2471:−
2461:∞
2452:∫
2427:
2346:ω
2324:ω
2304:ω
2292:∞
2284:∞
2281:−
2277:∫
2269:π
2212:→
2122:ω
2116:
2082:ω
2076:
2055:∞
2040:∑
1823:×
1819:∇
1790:⋅
1787:∇
1757:∇
1679:∫
1547:‖
1544:
1538:‖
1535:⋅
1532:‖
1529:
1523:‖
1520:≤
1517:‖
1514:
1508:
1502:‖
1463:‖
1454:‖
1445:≤
1436:‖
1427:
1421:‖
1400:‖
1397:
1391:‖
1347:‖
1338:‖
1329:≤
1320:‖
1311:
1305:‖
1192:sequences
1118:bijective
1028:
877:∈
819:
807:≡
731:
694:
670:
641:→
522:…
468:…
404:morphisms
348:
342:β
331:
325:α
309:β
298:α
290:
258:→
76:when the
2649:Function
2643:See also
2635: .
2231:integral
1939:variance
1864:geometry
1858:Isometry
1848:Geometry
1746:gradient
1724:Gradient
1616:calculus
1591:Examples
408:matrices
368:for all
185:Operator
74:function
70:operator
62:function
54:operator
2780:Algebra
2709: .
2186:, ... )
1288:bounded
1216:spectra
1210:form a
624:), and
239:mapping
58:mapping
2757:
2732:
2685:
2585:. ...
2508:
2500:
2485:
2477:
2380:Given
2343:
2335:
2327:
2321:
2310:
2233:form:
2131:
2125:
2091:
2085:
2060:
2027:
2014:
1941:, and
1894:do not
1876:groups
1744:Grad (
1730:, and
1640:
1632:
1451:
1412:
1335:
1295:> 0
1234:, and
1175:, and
1017:, and
769:
664:
450:
356:
276:linear
217:. Let
78:domain
2697:into
1773:Div (
1276:norms
784:Then
231:field
195:(see
66:space
52:, an
2755:ISBN
2730:ISBN
2683:ISBN
2603:(6)
2596:(2)
2373:The
1969:and
1856:and
1808:Curl
1732:Curl
1604:and
1379:and
1244:and
1240:Let
1165:rank
1112:-by-
499:and
434:and
374:and
236:. A
221:and
169:and
2705:on
2581:to
2550:on
2542:to
2113:sin
2073:cos
1862:In
1702:.
1406:inf
1371:in
1282:to
1160:.
1140:to
1053:if
549:in
495:in
394:in
380:in
279:if
273:is
225:be
199:).
132:to
60:or
48:In
41:or
2776::
2716:^
2691:.
2386:=
2179:,
2172:,
2165:,
2158:,
2151:(
1937:,
1740::
1726:,
1722:,
1718:,
1714:,
1614:,
1587:.
1383::
1230:,
1203:.
1179:.
1171:,
1167:,
398:.
389:,
177:,
2763:.
2738:.
2707:X
2699:X
2695:X
2633:V
2583:W
2579:V
2567:)
2565:V
2563:(
2561:ℒ
2556:V
2552:V
2544:V
2540:V
2511:t
2504:d
2497:)
2494:t
2491:(
2488:f
2480:t
2474:s
2467:e
2456:0
2448:=
2445:)
2442:s
2439:(
2436:}
2433:f
2430:{
2422:L
2417:=
2414:)
2411:s
2408:(
2405:F
2394:)
2392:s
2390:(
2388:f
2384:f
2339:d
2330:t
2318:i
2314:e
2307:)
2301:(
2298:g
2289:+
2266:2
2261:1
2256:=
2253:)
2250:t
2247:(
2244:f
2216:C
2208:R
2192:ℓ
2184:2
2181:b
2177:2
2174:a
2170:1
2167:b
2163:1
2160:a
2156:0
2153:a
2137:)
2134:t
2128:n
2119:(
2108:n
2104:b
2100:+
2097:)
2094:t
2088:n
2079:(
2068:n
2064:a
2050:1
2047:=
2044:n
2036:+
2031:2
2022:0
2018:a
2008:=
2005:)
2002:t
1999:(
1996:f
1688:t
1683:0
1650:t
1646:d
1636:d
1559:.
1541:B
1526:A
1511:B
1505:A
1489:U
1475:.
1472:}
1467:U
1458:x
1448:c
1440:V
1431:x
1424:A
1418::
1415:c
1409:{
1403:=
1394:A
1381:V
1377:U
1373:U
1368:x
1351:U
1342:x
1332:c
1324:V
1315:x
1308:A
1293:c
1284:V
1280:U
1261:R
1246:V
1242:U
1148:V
1128:U
1114:m
1110:n
1094:j
1090:y
1086:=
1081:i
1077:x
1071:j
1066:i
1062:a
1040:y
1036:=
1032:x
1025:A
1005:x
983:j
978:i
974:a
951:n
946:1
943:=
940:i
936:}
930:i
925:u
920:{
900:A
880:K
872:j
867:i
863:a
840:j
835:)
829:i
824:u
816:A
812:(
802:j
797:i
793:a
772:.
764:j
759:v
752:j
747:)
741:i
736:u
728:A
724:(
717:i
713:x
709:=
704:i
699:u
691:A
686:i
682:x
678:=
674:x
667:A
644:V
638:U
635::
632:A
608:U
586:i
581:u
574:i
570:x
566:=
562:x
551:V
535:m
530:v
525:,
519:,
514:1
509:v
497:U
481:n
476:u
471:,
465:,
460:1
455:u
440:K
436:V
422:U
412:K
396:K
391:β
387:α
382:U
377:y
371:x
352:y
345:A
339:+
335:x
328:A
322:=
318:)
313:y
306:+
302:x
294:(
287:A
261:V
255:U
252::
249:A
234:K
223:V
219:U
147:n
142:R
118:n
113:R
45:.
34:.
20:)
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