4854:
3018:
In finite dimensions, this is equivalent to the pairing being nondegenerate (the spaces necessarily having the same dimensions). For modules (instead of vector spaces), just as how a nondegenerate form is weaker than a unimodular form, a nondegenerate pairing is a weaker notion than a perfect
2315:
2675:
and is independent of the antisymmetric part. In this case there is a one-to-one correspondence between the symmetric part of the bilinear form and the quadratic form, and it makes sense to speak of the symmetric bilinear form associated with a quadratic form.
631:
2469:
2001:
2575:
2157:
is reflexive if and only if it is either symmetric or alternating. In the absence of reflexivity we have to distinguish left and right orthogonality. In a reflexive space the left and right radicals agree and are termed the
745:
1194:
is an isomorphism. Given a finitely generated module over a commutative ring, the pairing may be injective (hence "nondegenerate" in the above sense) but not unimodular. For example, over the integers, the pairing
3215:
2225:
506:
2400:
1887:
1593:). These statements are independent of the chosen basis. For a module over a commutative ring, a unimodular form is one for which the determinant of the associate matrix is a
299:
4743:
2512:
4225:
1137:
1061:
1841:). A bilinear form is alternating if and only if its coordinate matrix is skew-symmetric and the diagonal entries are all zero (which follows from skew-symmetry when
778:
1163:
1087:
4406:
666:
1597:(for example 1), hence the term; note that a form whose matrix determinant is non-zero but not a unit will be nondegenerate but not unimodular, for example
3112:
1460:, this is equivalent to the condition that the left and equivalently right radicals be trivial. For finite-dimensional spaces, this is often taken as the
4569:
1019:. More concretely, for a finite-dimensional vector space, non-degenerate means that every non-zero element pairs non-trivially with some other element:
1589:
of the associated matrix is zero. Likewise, a nondegenerate form is one for which the determinant of the associated matrix is non-zero (the matrix is
4696:
4551:
4527:
3007:. It may happen that these mappings are isomorphisms; assuming finite dimensions, if one is an isomorphism, the other must be. When this occurs,
4297:
4270:
4208:
4131:
4102:
4084:
4419:
4119:
4508:
4399:
4337:
4315:
4237:
4161:
4042:
1826:
then a skew-symmetric form is the same as a symmetric form and there exist symmetric/skew-symmetric forms that are not alternating.
4289:
4778:
4423:
635:
A bilinear form has different matrices on different bases. However, the matrices of a bilinear form on different bases are all
2310:{\displaystyle W^{\perp }=\left\{\mathbf {v} \mid B(\mathbf {v} ,\mathbf {w} )=0{\text{ for all }}\mathbf {w} \in W\right\}.}
4574:
4363:
4329:
4030:
4630:
4906:
4857:
4579:
4564:
4392:
3894:
4594:
626:{\displaystyle B(\mathbf {x} ,\mathbf {y} )=\mathbf {x} ^{\textsf {T}}A\mathbf {y} =\sum _{i,j=1}^{n}x_{i}A_{ij}y_{j}.}
4358:
4176:
4839:
4599:
4793:
4717:
4262:
4181:
4834:
4891:
4650:
1260:
4584:
1457:
4686:
4487:
1811:
810:
363:
4559:
4901:
4896:
4783:
3073:
1628:
4814:
4758:
4722:
3859:
1578:
367:
2464:{\displaystyle B(\mathbf {u} ,\mathbf {v} )\leq C\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}
2219:
1838:
1393:
4283:
1996:{\displaystyle B^{+}={\tfrac {1}{2}}(B+{}^{\text{t}}B)\qquad B^{-}={\tfrac {1}{2}}(B-{}^{\text{t}}B),}
275:
4797:
3337:
305:
77:
64:
4763:
4701:
4415:
4353:
3864:
2363:
1590:
317:
69:
4875:
4788:
4655:
3330:
2704:
1594:
1574:
636:
309:
2166:
of the bilinear form: the subspace of all vectors orthogonal with every other vector. A vector
1818:
is not 2 then the converse is also true: every skew-symmetric form is alternating. However, if
1385:
respectively; they are the vectors orthogonal to the whole space on the left and on the right.
4768:
4333:
4311:
4293:
4266:
4233:
4204:
4188:
4157:
4127:
4098:
4080:
4038:
3889:
3081:
3077:
2489:
1101:
1025:
977:
781:
337:
4171:
1884:
then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows
750:
4773:
4691:
4660:
4640:
4625:
4620:
4615:
4452:
4279:
4243:
4137:
4059:
4048:
3874:
2961:
1834:
1679:
1364:
1142:
1066:
341:
4635:
4589:
4537:
4532:
4503:
4384:
4254:
4247:
4229:
4141:
4123:
4052:
4034:
3580:
3293:
3062:
2897:
1016:
4462:
1176:
The corresponding notion for a module over a commutative ring is that a bilinear form is
2695:, this correspondence between quadratic forms and symmetric bilinear forms breaks down.
4824:
4676:
4477:
4153:
4111:
3884:
2918:
2708:
2609:
2585:
1830:
328:
3019:
pairing. A pairing can be nondegenerate without being a perfect pairing, for instance
4885:
4829:
4753:
4482:
4467:
4457:
3920:
3899:
4819:
4472:
4442:
4193:
3854:
3065:
discusses "eight types of inner product". To define them he uses diagonal matrices
1876:
are equal, and skew-symmetric if and only if they are negatives of one another. If
56:
39:
2570:{\displaystyle B(\mathbf {u} ,\mathbf {u} )\geq c\left\|\mathbf {u} \right\|^{2}.}
976:
where the dot ( ⋅ ) indicates the slot into which the argument for the resulting
4748:
4738:
4645:
4447:
4217:
3879:
3869:
3835:
3349:
3254:
Some of the real symmetric cases are very important. The positive definite case
3101:
1586:
269:
31:
4148:
Harvey, F. Reese (1990), "Chapter 2: The Eight Types of Inner
Product Spaces",
17:
4871:
4681:
4521:
4517:
4513:
4376:
2846:
830:
313:
100:
3072:
having only +1 or −1 for non-zero elements. Some of the "inner products" are
4076:
2330:
2210:
is nonsingular, and thus if and only if the bilinear form is nondegenerate.
981:
740:{\displaystyle \mathbf {f} _{j}=\sum _{i=1}^{n}S_{i,j}\mathbf {e} _{i},}
4371:
2784:
is a linear map the corresponding bilinear form is given by composing
3210:{\displaystyle \sum _{k=1}^{p}x_{k}y_{k}-\sum _{k=p+1}^{n}x_{k}y_{k}}
2888:
Likewise, symmetric bilinear forms may be thought of as elements of
3250:. Then he articulates the connection to traditional terminology:
2178:, is in the radical of a bilinear form with matrix representation
2319:
For a non-degenerate form on a finite-dimensional space, the map
2711:, there is a canonical correspondence between bilinear forms on
4388:
3061:
Terminology varies in coverage of bilinear forms. For example,
2964:
from two vector spaces over the same base field to that field
3048:
is nondegenerate, but induces multiplication by 2 on the map
1213:
is nondegenerate but not unimodular, as the induced map from
304:
The definition of a bilinear form can be extended to include
4071:
Cooperstein, Bruce (2010), "Ch 8: Bilinear Forms and Maps",
787:. Then, the matrix of the bilinear form on the new basis is
4261:, Cambridge Studies in Advanced Mathematics, vol. 50,
1829:
A bilinear form is symmetric (respectively skew-symmetric)
4326:
Principal
Structures and Methods of Representation Theory
2671:
is determined by the symmetric part of the bilinear form
1009:
is an isomorphism, then both are, and the bilinear form
4306:
Shilov, Georgi E. (1977), Silverman, Richard A. (ed.),
4870:
This article incorporates material from
Unimodular on
3921:"Chapter 3. Bilinear forms — Lecture notes for MA1212"
1955:
1905:
3115:
2859:, so bilinear forms may be thought of as elements of
2515:
2403:
2228:
1890:
1852:
A bilinear form is symmetric if and only if the maps
1145:
1104:
1069:
1028:
753:
669:
509:
278:
2875:
is finite-dimensional) is canonically isomorphic to
4807:
4731:
4710:
4669:
4608:
4550:
4496:
4431:
4203:, vol. I (2nd ed.), Courier Corporation,
1585:, a bilinear form is degenerate if and only if the
4744:Spectral theory of ordinary differential equations
3209:
2569:
2463:
2309:
1995:
1157:
1131:
1081:
1055:
772:
739:
625:
293:
4025:Adkins, William A.; Weintraub, Steven H. (1992),
1833:its coordinate matrix (relative to any basis) is
81:). In other words, a bilinear form is a function
4876:Creative Commons Attribution/Share-Alike License
4226:Ergebnisse der Mathematik und ihrer Grenzgebiete
2987:Here we still have induced linear mappings from
2906:) and alternating bilinear forms as elements of
1619:Symmetric, skew-symmetric, and alternating forms
3972:
3252:
2019:Reflexive bilinear forms and orthogonal vectors
1560:This form will be nondegenerate if and only if
484:with respect to this basis, and similarly, the
340:, which are similar to bilinear forms but are
4400:
8:
1240:is finite-dimensional then one can identify
4328:, Translations of Mathematical Monographs,
2204:. It is trivial if and only if the matrix
4435:
4407:
4393:
4385:
4259:Clifford Algebras and the Classical Groups
3948:
3201:
3191:
3181:
3164:
3151:
3141:
3131:
3120:
3114:
2739:the corresponding linear map is given by
2558:
2549:
2530:
2522:
2514:
2476:A bilinear form on a normed vector space
2449:
2436:
2418:
2410:
2402:
2288:
2283:
2269:
2261:
2247:
2233:
2227:
1978:
1976:
1954:
1945:
1928:
1926:
1904:
1895:
1889:
1775:Every alternating form is skew-symmetric.
1144:
1103:
1068:
1027:
758:
752:
728:
723:
710:
700:
689:
676:
671:
668:
614:
601:
591:
581:
564:
552:
543:
542:
541:
536:
524:
516:
508:
285:
281:
280:
277:
4697:Group algebra of a locally compact group
4008:
3936:
3912:
3291:, then Lorentzian space is also called
4324:Zhelobenko, Dmitriĭ Petrovich (2006),
4027:Algebra: An Approach via Module Theory
3996:
3984:
2960:Much of the theory is available for a
2198:. The radical is always a subspace of
987:For a finite-dimensional vector space
3960:
7:
3269:, while the case of a single minus,
3109:are spelled out. The bilinear form
2355:Bounded and elliptic bilinear forms
825:defines a pair of linear maps from
4120:Undergraduate Texts in Mathematics
3088:, the instances with real numbers
336:, one is often more interested in
301:is an example of a bilinear form.
75:(the elements of which are called
62:(the elements of which are called
25:
2586:Quadratic form § Definitions
1320:to be the bilinear form given by
4853:
4852:
4779:Topological quantum field theory
4116:Finite-dimensional vector spaces
2550:
2531:
2523:
2450:
2437:
2419:
2411:
2289:
2270:
2262:
2248:
1623:We define a bilinear form to be
724:
672:
553:
537:
525:
517:
294:{\displaystyle \mathbb {R} ^{n}}
2956:Pairs of distinct vector spaces
1940:
1523:one can obtain a bilinear form
1392:is finite-dimensional then the
27:Scalar-valued bilinear function
4874:, which is licensed under the
3084:. Rather than a general field
2554:
2546:
2535:
2519:
2454:
2446:
2441:
2433:
2423:
2407:
2274:
2258:
2114:be a reflexive bilinear form.
1987:
1966:
1937:
1916:
1781:This can be seen by expanding
1120:
1108:
1044:
1032:
529:
513:
1:
4575:Uniform boundedness principle
4330:American Mathematical Society
4031:Graduate Texts in Mathematics
2608:, there exists an associated
2172:, with matrix representation
1440:are linear isomorphisms from
1414:. If this number is equal to
1276:is infinite-dimensional then
805:Non-degenerate bilinear forms
103:in each argument separately:
3895:System of bilinear equations
3736:Conversely, a bilinear form
4359:Encyclopedia of Mathematics
4285:Linear Algebra and Geometry
4177:Encyclopedia of Mathematics
4174:, in Hazewinkel, M. (ed.),
3973:Adkins & Weintraub 1992
3316:will be referred to as the
2831:The set of all linear maps
2766:In the other direction, if
2699:Relation to tensor products
2217:is a subspace. Define the
2128:orthogonal with respect to
1294:restricted to the image of
440:matrix of the bilinear form
4923:
4718:Invariant subspace problem
4263:Cambridge University Press
4182:Kluwer Academic Publishers
3694:induces the bilinear form
3628:induces the bilinear form
2583:
808:
497:represents another vector
4848:
4438:
4220:; Husemoller, D. (1973),
4199:Jacobson, Nathan (2009),
2580:Associated quadratic form
2492:, if there is a constant
2377:, if there is a constant
1456:is nondegenerate. By the
1250:. One can then show that
927:This is often denoted as
348:Coordinate representation
4687:Spectrum of a C*-algebra
4282:; A. O. Remizov (2012),
4222:Symmetric Bilinear Forms
4150:Spinors and calibrations
4093:Grove, Larry C. (1997),
1405:is equal to the rank of
1233:is multiplication by 2.
1132:{\displaystyle B(x,y)=0}
1056:{\displaystyle B(x,y)=0}
811:Degenerate bilinear form
4784:Noncommutative geometry
4073:Advanced Linear Algebra
3586:canonical bilinear form
2333:, and the dimension of
1577:then, relative to some
773:{\displaystyle S_{i,j}}
4840:Tomita–Takesaki theory
4815:Approximation property
4759:Calculus of variations
4097:, Wiley-Interscience,
3860:Category:Bilinear maps
3322:
3211:
3186:
3136:
2788:with the bilinear map
2735:is a bilinear form on
2590:For any bilinear form
2571:
2465:
2311:
1997:
1159:
1158:{\displaystyle x\in V}
1133:
1083:
1082:{\displaystyle y\in V}
1057:
774:
741:
705:
627:
586:
295:
4835:Banach–Mazur distance
4798:Generalized functions
4170:Popov, V. L. (1987),
4095:Groups and characters
3212:
3160:
3116:
2667:, the quadratic form
2584:Further information:
2572:
2466:
2362:A bilinear form on a
2312:
2220:orthogonal complement
1998:
1509:Given any linear map
1244:with its double dual
1160:
1134:
1084:
1058:
980:is to be placed (see
809:Further information:
775:
742:
685:
639:. More precisely, if
628:
560:
296:
4580:Kakutani fixed-point
4565:Riesz representation
4122:, Berlin, New York:
3113:
2896:(dual of the second
2513:
2401:
2226:
2009:is the transpose of
1888:
1458:rank–nullity theorem
1285:is the transpose of
1143:
1102:
1067:
1026:
815:Every bilinear form
751:
667:
659:is another basis of
507:
478:represents a vector
318:module homomorphisms
276:
4907:Multilinear algebra
4764:Functional calculus
4723:Mahler's conjecture
4702:Von Neumann algebra
4416:Functional analysis
4079:, pp. 249–88,
3865:Inner product space
3667:, and a linear map
3301:. The special case
3299:Minkowski spacetime
3219:real symmetric case
2364:normed vector space
2285: for all
1615:over the integers.
1566:is an isomorphism.
1310:one can define the
4789:Riemann hypothesis
4488:Topological vector
4280:Shafarevich, I. R.
4184:, pp. 390–392
4156:, pp. 19–40,
3584:, also called the
3207:
3094:, complex numbers
3078:sesquilinear forms
2705:universal property
2567:
2499:such that for all
2461:
2383:such that for all
2307:
1993:
1964:
1914:
1575:finite-dimensional
1464:of nondegeneracy:
1263:of the linear map
1155:
1129:
1079:
1053:
770:
737:
623:
366:vector space with
338:sesquilinear forms
291:
234:
230:
154:
150:
4866:
4865:
4769:Integral operator
4546:
4545:
4299:978-3-642-30993-9
4272:978-0-521-55177-9
4210:978-0-486-47189-1
4133:978-0-387-90093-3
4104:978-0-471-16340-4
4086:978-1-4398-2966-0
4033:, vol. 136,
3890:Sesquilinear form
2286:
2184:, if and only if
2015:(defined above).
1981:
1963:
1931:
1913:
978:linear functional
782:invertible matrix
545:
344:in one argument.
16:(Redirected from
4914:
4892:Abstract algebra
4856:
4855:
4774:Jones polynomial
4692:Operator algebra
4436:
4409:
4402:
4395:
4386:
4381:
4367:
4342:
4320:
4302:
4275:
4255:Porteous, Ian R.
4250:
4228:, vol. 73,
4213:
4185:
4166:
4144:
4107:
4089:
4067:
4055:
4012:
4006:
4000:
3994:
3988:
3982:
3976:
3970:
3964:
3958:
3952:
3946:
3940:
3934:
3928:
3927:
3925:
3917:
3875:Multilinear form
3843:
3833:
3827:
3792:
3753:
3732:
3693:
3666:
3627:
3597:
3578:is known as the
3577:
3542:
3528:
3514:
3498:
3467:
3423:
3374:
3356:
3347:
3340:
3335:
3315:
3290:
3282:Lorentzian space
3279:
3264:
3249:
3235:
3216:
3214:
3213:
3208:
3206:
3205:
3196:
3195:
3185:
3180:
3156:
3155:
3146:
3145:
3135:
3130:
3108:
3099:
3093:
3087:
3074:symplectic forms
3057:
3047:
3032:
3011:is said to be a
3006:
3000:
2996:
2990:
2983:
2962:bilinear mapping
2946:
2934:
2926:
2916:
2905:
2895:
2884:
2874:
2870:
2858:
2844:
2827:
2817:
2805:
2783:
2763:
2738:
2734:
2728:
2715:and linear maps
2714:
2694:
2686:
2666:
2655:
2624:
2607:
2576:
2574:
2573:
2568:
2563:
2562:
2557:
2553:
2534:
2526:
2508:
2498:
2483:
2470:
2468:
2467:
2462:
2457:
2453:
2444:
2440:
2422:
2414:
2396:
2382:
2372:
2350:
2338:
2328:
2316:
2314:
2313:
2308:
2303:
2299:
2292:
2287:
2284:
2273:
2265:
2251:
2238:
2237:
2216:
2209:
2203:
2197:
2183:
2177:
2171:
2156:
2151:A bilinear form
2147:
2113:
2077:
2062:
2043:
2026:A bilinear form
2014:
2008:
2002:
2000:
1999:
1994:
1983:
1982:
1979:
1977:
1965:
1956:
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1949:
1933:
1932:
1929:
1927:
1915:
1906:
1900:
1899:
1883:
1875:
1848:
1825:
1817:
1803:
1768:
1764:
1758:
1752:
1724:
1723:
1716:
1715:
1707:
1703:
1697:
1674:
1670:
1664:
1658:
1614:
1584:
1572:
1565:
1522:
1505:
1491:
1455:
1450:. In this case
1449:
1443:
1439:
1430:
1421:
1413:
1404:
1391:
1384:
1375:
1362:
1319:
1309:
1303:
1297:
1293:
1284:
1275:
1271:
1258:
1249:
1243:
1239:
1232:
1222:
1212:
1193:
1182:
1181:
1171:
1164:
1162:
1161:
1156:
1138:
1136:
1135:
1130:
1095:
1088:
1086:
1085:
1080:
1062:
1060:
1059:
1054:
1014:
1008:
999:
990:
973:
950:
924:
893:
861:
837:
828:
824:
820:
795:
786:
779:
777:
776:
771:
769:
768:
746:
744:
743:
738:
733:
732:
727:
721:
720:
704:
699:
681:
680:
675:
662:
658:
632:
630:
629:
624:
619:
618:
609:
608:
596:
595:
585:
580:
556:
548:
547:
546:
540:
528:
520:
502:
496:
490:
483:
477:
471:
461:
437:
401:
388:
361:
357:
342:conjugate linear
335:
327:is the field of
326:
300:
298:
297:
292:
290:
289:
284:
264:
235:
231:
228:
184:
155:
151:
148:
98:
61:
54:
21:
4922:
4921:
4917:
4916:
4915:
4913:
4912:
4911:
4882:
4881:
4867:
4862:
4844:
4808:Advanced topics
4803:
4727:
4706:
4665:
4631:Hilbert–Schmidt
4604:
4595:Gelfand–Naimark
4542:
4492:
4427:
4413:
4372:"Bilinear form"
4370:
4354:"Bilinear form"
4352:
4349:
4340:
4323:
4318:
4305:
4300:
4278:
4273:
4253:
4240:
4230:Springer-Verlag
4216:
4211:
4198:
4180:, vol. 1,
4172:"Bilinear form"
4169:
4164:
4147:
4134:
4124:Springer-Verlag
4112:Halmos, Paul R.
4110:
4105:
4092:
4087:
4070:
4058:
4045:
4035:Springer-Verlag
4024:
4021:
4016:
4015:
4007:
4003:
3995:
3991:
3983:
3979:
3971:
3967:
3959:
3955:
3949:Zhelobenko 2006
3947:
3943:
3935:
3931:
3923:
3919:
3918:
3914:
3909:
3904:
3850:
3839:
3829:
3794:
3759:
3737:
3695:
3668:
3629:
3602:
3589:
3581:natural pairing
3547:
3530:
3516:
3502:
3499:
3470:
3468:
3426:
3424:
3382:
3358:
3352:
3343:
3338:
3333:
3327:
3325:General modules
3302:
3294:Minkowski space
3285:
3270:
3267:Euclidean space
3255:
3237:
3222:
3197:
3187:
3147:
3137:
3111:
3110:
3104:
3095:
3089:
3085:
3082:Hermitian forms
3070:
3063:F. Reese Harvey
3049:
3034:
3020:
3013:perfect pairing
3002:
2998:
2992:
2988:
2985:
2967:
2958:
2953:
2951:Generalizations
2936:
2928:
2922:
2907:
2901:
2898:symmetric power
2889:
2876:
2872:
2860:
2850:
2832:
2819:
2807:
2789:
2767:
2764:
2742:
2736:
2730:
2716:
2712:
2701:
2688:
2680:
2660:
2626:
2612:
2591:
2588:
2582:
2545:
2544:
2511:
2510:
2500:
2493:
2477:
2445:
2432:
2399:
2398:
2384:
2378:
2366:
2357:
2340:
2334:
2320:
2246:
2242:
2229:
2224:
2223:
2214:
2205:
2199:
2185:
2179:
2173:
2167:
2152:
2149:
2134:
2097:
2091:
2064:
2049:
2027:
2021:
2010:
2004:
1975:
1941:
1925:
1891:
1886:
1885:
1877:
1866:
1859:
1853:
1842:
1819:
1815:
1782:
1766:
1760:
1754:
1727:
1721:
1720:
1713:
1712:
1705:
1699:
1684:
1672:
1666:
1660:
1633:
1621:
1598:
1582:
1570:
1561:
1558:
1510:
1507:
1497:
1478:
1451:
1445:
1441:
1438:
1432:
1429:
1423:
1415:
1412:
1406:
1403:
1397:
1389:
1383:
1377:
1374:
1368:
1358:
1347:
1315:
1305:
1299:
1295:
1292:
1286:
1283:
1277:
1273:
1270:
1264:
1257:
1251:
1245:
1241:
1237:
1224:
1214:
1196:
1185:
1179:
1178:
1166:
1141:
1140:
1100:
1099:
1090:
1065:
1064:
1024:
1023:
1010:
1007:
1001:
998:
992:
991:, if either of
988:
974:
959:
953:
951:
936:
930:
925:
902:
896:
894:
871:
865:
852:
845:
839:
833:
826:
822:
816:
813:
807:
802:
788:
784:
754:
749:
748:
722:
706:
670:
665:
664:
660:
656:
647:
640:
610:
597:
587:
535:
505:
504:
498:
492:
485:
479:
473:
466:
459:
450:
443:
435:
426:
412:
407:
393:
386:
377:
370:
359:
353:
350:
331:
329:complex numbers
324:
279:
274:
273:
236:
233:
229:
187:
156:
153:
149:
107:
82:
59:
42:
28:
23:
22:
18:Unimodular form
15:
12:
11:
5:
4920:
4918:
4910:
4909:
4904:
4902:Linear algebra
4899:
4897:Bilinear forms
4894:
4884:
4883:
4864:
4863:
4861:
4860:
4849:
4846:
4845:
4843:
4842:
4837:
4832:
4827:
4825:Choquet theory
4822:
4817:
4811:
4809:
4805:
4804:
4802:
4801:
4791:
4786:
4781:
4776:
4771:
4766:
4761:
4756:
4751:
4746:
4741:
4735:
4733:
4729:
4728:
4726:
4725:
4720:
4714:
4712:
4708:
4707:
4705:
4704:
4699:
4694:
4689:
4684:
4679:
4677:Banach algebra
4673:
4671:
4667:
4666:
4664:
4663:
4658:
4653:
4648:
4643:
4638:
4633:
4628:
4623:
4618:
4612:
4610:
4606:
4605:
4603:
4602:
4600:Banach–Alaoglu
4597:
4592:
4587:
4582:
4577:
4572:
4567:
4562:
4556:
4554:
4548:
4547:
4544:
4543:
4541:
4540:
4535:
4530:
4528:Locally convex
4525:
4511:
4506:
4500:
4498:
4494:
4493:
4491:
4490:
4485:
4480:
4475:
4470:
4465:
4460:
4455:
4450:
4445:
4439:
4433:
4429:
4428:
4414:
4412:
4411:
4404:
4397:
4389:
4383:
4382:
4368:
4348:
4347:External links
4345:
4344:
4343:
4338:
4321:
4316:
4308:Linear Algebra
4303:
4298:
4276:
4271:
4251:
4238:
4214:
4209:
4196:
4167:
4162:
4154:Academic Press
4145:
4132:
4108:
4103:
4090:
4085:
4068:
4056:
4043:
4020:
4017:
4014:
4013:
4011:, p. 233.
4001:
3989:
3977:
3975:, p. 359.
3965:
3953:
3941:
3939:, p. 346.
3929:
3911:
3910:
3908:
3905:
3903:
3902:
3897:
3892:
3887:
3885:Quadratic form
3882:
3877:
3872:
3867:
3862:
3857:
3851:
3849:
3846:
3469:
3425:
3381:
3326:
3323:
3217:is called the
3204:
3200:
3194:
3190:
3184:
3179:
3176:
3173:
3170:
3167:
3163:
3159:
3154:
3150:
3144:
3140:
3134:
3129:
3126:
3123:
3119:
3068:
2966:
2957:
2954:
2952:
2949:
2919:exterior power
2741:
2709:tensor product
2700:
2697:
2610:quadratic form
2581:
2578:
2566:
2561:
2556:
2552:
2548:
2543:
2540:
2537:
2533:
2529:
2525:
2521:
2518:
2460:
2456:
2452:
2448:
2443:
2439:
2435:
2431:
2428:
2425:
2421:
2417:
2413:
2409:
2406:
2356:
2353:
2306:
2302:
2298:
2295:
2291:
2282:
2279:
2276:
2272:
2268:
2264:
2260:
2257:
2254:
2250:
2245:
2241:
2236:
2232:
2092:
2022:
2020:
2017:
1992:
1989:
1986:
1974:
1971:
1968:
1962:
1959:
1953:
1948:
1944:
1939:
1936:
1924:
1921:
1918:
1912:
1909:
1903:
1898:
1894:
1864:
1857:
1839:skew-symmetric
1837:(respectively
1831:if and only if
1812:characteristic
1808:
1807:
1806:
1805:
1779:
1776:
1773:
1714:skew-symmetric
1709:
1676:
1620:
1617:
1533:
1466:
1436:
1427:
1410:
1401:
1381:
1372:
1322:
1290:
1281:
1268:
1255:
1174:
1173:
1154:
1151:
1148:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1097:
1078:
1075:
1072:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1015:is said to be
1005:
996:
957:
952:
934:
929:
900:
895:
869:
864:
850:
843:
806:
803:
801:
798:
767:
764:
761:
757:
736:
731:
726:
719:
716:
713:
709:
703:
698:
695:
692:
688:
684:
679:
674:
652:
645:
622:
617:
613:
607:
604:
600:
594:
590:
584:
579:
576:
573:
570:
567:
563:
559:
555:
551:
539:
534:
531:
527:
523:
519:
515:
512:
455:
448:
438:is called the
431:
422:
410:
382:
375:
349:
346:
288:
283:
266:
265:
185:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4919:
4908:
4905:
4903:
4900:
4898:
4895:
4893:
4890:
4889:
4887:
4880:
4879:
4877:
4873:
4859:
4851:
4850:
4847:
4841:
4838:
4836:
4833:
4831:
4830:Weak topology
4828:
4826:
4823:
4821:
4818:
4816:
4813:
4812:
4810:
4806:
4799:
4795:
4792:
4790:
4787:
4785:
4782:
4780:
4777:
4775:
4772:
4770:
4767:
4765:
4762:
4760:
4757:
4755:
4754:Index theorem
4752:
4750:
4747:
4745:
4742:
4740:
4737:
4736:
4734:
4730:
4724:
4721:
4719:
4716:
4715:
4713:
4711:Open problems
4709:
4703:
4700:
4698:
4695:
4693:
4690:
4688:
4685:
4683:
4680:
4678:
4675:
4674:
4672:
4668:
4662:
4659:
4657:
4654:
4652:
4649:
4647:
4644:
4642:
4639:
4637:
4634:
4632:
4629:
4627:
4624:
4622:
4619:
4617:
4614:
4613:
4611:
4607:
4601:
4598:
4596:
4593:
4591:
4588:
4586:
4583:
4581:
4578:
4576:
4573:
4571:
4568:
4566:
4563:
4561:
4558:
4557:
4555:
4553:
4549:
4539:
4536:
4534:
4531:
4529:
4526:
4523:
4519:
4515:
4512:
4510:
4507:
4505:
4502:
4501:
4499:
4495:
4489:
4486:
4484:
4481:
4479:
4476:
4474:
4471:
4469:
4466:
4464:
4461:
4459:
4456:
4454:
4451:
4449:
4446:
4444:
4441:
4440:
4437:
4434:
4430:
4425:
4421:
4417:
4410:
4405:
4403:
4398:
4396:
4391:
4390:
4387:
4379:
4378:
4373:
4369:
4365:
4361:
4360:
4355:
4351:
4350:
4346:
4341:
4339:0-8218-3731-1
4335:
4331:
4327:
4322:
4319:
4317:0-486-63518-X
4313:
4309:
4304:
4301:
4295:
4291:
4287:
4286:
4281:
4277:
4274:
4268:
4264:
4260:
4256:
4252:
4249:
4245:
4241:
4239:3-540-06009-X
4235:
4231:
4227:
4223:
4219:
4215:
4212:
4206:
4202:
4201:Basic Algebra
4197:
4195:
4192:, p. 390, at
4191:
4190:
4189:Bilinear form
4183:
4179:
4178:
4173:
4168:
4165:
4163:0-12-329650-1
4159:
4155:
4151:
4146:
4143:
4139:
4135:
4129:
4125:
4121:
4117:
4113:
4109:
4106:
4100:
4096:
4091:
4088:
4082:
4078:
4074:
4069:
4065:
4061:
4057:
4054:
4050:
4046:
4044:3-540-97839-9
4040:
4036:
4032:
4028:
4023:
4022:
4018:
4010:
4009:Bourbaki 1970
4005:
4002:
3999:, p. 23.
3998:
3993:
3990:
3987:, p. 22.
3986:
3981:
3978:
3974:
3969:
3966:
3962:
3957:
3954:
3951:, p. 11.
3950:
3945:
3942:
3938:
3937:Jacobson 2009
3933:
3930:
3926:. 2021-01-16.
3922:
3916:
3913:
3906:
3901:
3900:Metric tensor
3898:
3896:
3893:
3891:
3888:
3886:
3883:
3881:
3878:
3876:
3873:
3871:
3868:
3866:
3863:
3861:
3858:
3856:
3853:
3852:
3847:
3845:
3842:
3837:
3832:
3825:
3821:
3817:
3813:
3809:
3805:
3801:
3797:
3790:
3786:
3782:
3778:
3774:
3770:
3766:
3762:
3758:-linear maps
3757:
3752:
3748:
3744:
3740:
3734:
3730:
3726:
3722:
3718:
3714:
3710:
3706:
3702:
3698:
3691:
3687:
3683:
3679:
3675:
3671:
3664:
3660:
3656:
3652:
3648:
3644:
3640:
3636:
3632:
3625:
3621:
3617:
3613:
3609:
3605:
3601:A linear map
3599:
3596:
3592:
3587:
3583:
3582:
3575:
3571:
3567:
3563:
3559:
3555:
3551:
3548:⟨⋅,⋅⟩ :
3544:
3541:
3537:
3533:
3527:
3523:
3519:
3513:
3509:
3505:
3497:
3493:
3489:
3485:
3481:
3477:
3473:
3465:
3461:
3457:
3453:
3449:
3445:
3441:
3437:
3433:
3429:
3421:
3417:
3413:
3409:
3405:
3401:
3397:
3393:
3389:
3385:
3380:
3378:
3377:bilinear form
3373:
3369:
3365:
3361:
3355:
3351:
3346:
3342:
3332:
3324:
3321:
3319:
3313:
3309:
3305:
3300:
3296:
3295:
3288:
3283:
3277:
3273:
3268:
3262:
3258:
3251:
3248:
3244:
3240:
3233:
3229:
3225:
3220:
3202:
3198:
3192:
3188:
3182:
3177:
3174:
3171:
3168:
3165:
3161:
3157:
3152:
3148:
3142:
3138:
3132:
3127:
3124:
3121:
3117:
3107:
3103:
3098:
3092:
3083:
3079:
3076:and some are
3075:
3071:
3064:
3059:
3056:
3052:
3046:
3042:
3038:
3031:
3027:
3023:
3016:
3014:
3010:
3005:
2995:
2982:
2978:
2974:
2970:
2965:
2963:
2955:
2950:
2948:
2944:
2940:
2932:
2925:
2920:
2915:
2911:
2904:
2899:
2893:
2886:
2883:
2879:
2868:
2864:
2857:
2853:
2848:
2843:
2839:
2835:
2829:
2826:
2822:
2815:
2811:
2804:
2800:
2796:
2792:
2787:
2782:
2778:
2774:
2770:
2761:
2757:
2753:
2749:
2745:
2740:
2733:
2727:
2723:
2719:
2710:
2706:
2698:
2696:
2692:
2684:
2677:
2674:
2670:
2664:
2657:
2653:
2649:
2645:
2641:
2637:
2633:
2629:
2623:
2619:
2615:
2611:
2606:
2602:
2598:
2594:
2587:
2579:
2577:
2564:
2559:
2541:
2538:
2527:
2516:
2507:
2503:
2496:
2491:
2487:
2481:
2475:
2471:
2458:
2429:
2426:
2415:
2404:
2395:
2391:
2387:
2381:
2376:
2370:
2365:
2361:
2354:
2352:
2348:
2344:
2337:
2332:
2327:
2323:
2317:
2304:
2300:
2296:
2293:
2280:
2277:
2266:
2255:
2252:
2243:
2239:
2234:
2230:
2222:
2221:
2211:
2208:
2202:
2195:
2192:
2188:
2182:
2176:
2170:
2165:
2161:
2155:
2145:
2141:
2137:
2132:
2131:
2125:
2121:
2117:
2112:
2108:
2104:
2100:
2095:
2089:
2085:
2081:
2075:
2071:
2067:
2060:
2056:
2052:
2047:
2042:
2038:
2034:
2030:
2025:
2018:
2016:
2013:
2007:
1990:
1984:
1972:
1969:
1960:
1957:
1951:
1946:
1942:
1934:
1922:
1919:
1910:
1907:
1901:
1896:
1892:
1881:
1874:
1870:
1863:
1856:
1850:
1846:
1840:
1836:
1832:
1827:
1823:
1813:
1801:
1797:
1793:
1789:
1785:
1780:
1777:
1774:
1771:
1770:
1763:
1757:
1750:
1746:
1742:
1738:
1734:
1730:
1725:
1722:antisymmetric
1717:
1710:
1702:
1695:
1691:
1687:
1682:
1681:
1677:
1669:
1663:
1656:
1652:
1648:
1644:
1640:
1636:
1631:
1630:
1626:
1625:
1624:
1618:
1616:
1613:
1609:
1605:
1601:
1596:
1592:
1588:
1580:
1576:
1567:
1564:
1556:
1552:
1548:
1544:
1540:
1536:
1532:
1530:
1526:
1521:
1517:
1513:
1504:
1500:
1495:
1489:
1485:
1481:
1476:
1475:nondegenerate
1472:
1469:
1465:
1463:
1459:
1454:
1448:
1435:
1426:
1419:
1409:
1400:
1395:
1386:
1380:
1371:
1366:
1361:
1356:
1355:right radical
1352:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1318:
1313:
1308:
1302:
1289:
1280:
1267:
1262:
1254:
1248:
1234:
1231:
1227:
1221:
1217:
1211:
1207:
1203:
1199:
1192:
1188:
1183:
1169:
1165:implies that
1152:
1149:
1146:
1126:
1123:
1117:
1114:
1111:
1105:
1098:
1093:
1089:implies that
1076:
1073:
1070:
1050:
1047:
1041:
1038:
1035:
1029:
1022:
1021:
1020:
1018:
1017:nondegenerate
1013:
1004:
995:
985:
983:
979:
971:
967:
963:
956:
948:
944:
940:
933:
928:
922:
918:
914:
910:
906:
899:
891:
887:
883:
879:
875:
868:
863:
860:
856:
849:
842:
836:
832:
819:
812:
804:
799:
797:
794:
791:
783:
765:
762:
759:
755:
734:
729:
717:
714:
711:
707:
701:
696:
693:
690:
686:
682:
677:
655:
651:
644:
638:
633:
620:
615:
611:
605:
602:
598:
592:
588:
582:
577:
574:
571:
568:
565:
561:
557:
549:
532:
521:
510:
501:
495:
488:
482:
476:
469:
463:
458:
454:
447:
442:on the basis
441:
434:
430:
425:
421:
417:
413:
406:, defined by
405:
400:
396:
390:
385:
381:
374:
369:
365:
356:
347:
345:
343:
339:
334:
330:
321:
319:
315:
311:
307:
302:
286:
271:
262:
258:
254:
250:
247:
243:
239:
226:
222:
218:
214:
210:
206:
202:
198:
194:
190:
186:
182:
178:
174:
170:
166:
163:
159:
146:
142:
138:
134:
130:
126:
122:
118:
114:
110:
106:
105:
104:
102:
97:
93:
89:
85:
80:
79:
74:
71:
67:
66:
58:
53:
49:
45:
41:
37:
36:bilinear form
33:
19:
4869:
4868:
4820:Balanced set
4794:Distribution
4732:Applications
4585:Krein–Milman
4570:Closed graph
4375:
4357:
4325:
4307:
4284:
4258:
4221:
4200:
4194:Google Books
4187:
4175:
4149:
4115:
4094:
4072:
4063:
4060:Bourbaki, N.
4026:
4004:
3992:
3980:
3968:
3956:
3944:
3932:
3915:
3855:Bilinear map
3840:
3834:denotes the
3830:
3823:
3819:
3815:
3811:
3807:
3803:
3799:
3795:
3788:
3784:
3780:
3776:
3772:
3768:
3764:
3760:
3755:
3754:induces the
3750:
3746:
3742:
3738:
3735:
3728:
3724:
3720:
3716:
3712:
3708:
3704:
3700:
3696:
3689:
3685:
3681:
3677:
3673:
3669:
3662:
3658:
3654:
3650:
3646:
3642:
3638:
3634:
3630:
3623:
3619:
3615:
3611:
3607:
3603:
3600:
3594:
3590:
3585:
3579:
3573:
3569:
3565:
3561:
3557:
3553:
3549:
3546:The mapping
3545:
3539:
3535:
3531:
3525:
3521:
3517:
3511:
3507:
3503:
3500:
3495:
3491:
3487:
3483:
3479:
3475:
3471:
3463:
3459:
3455:
3451:
3447:
3443:
3439:
3435:
3431:
3427:
3419:
3415:
3411:
3407:
3403:
3399:
3395:
3391:
3387:
3383:
3376:
3375:is called a
3371:
3367:
3363:
3359:
3357:, a mapping
3353:
3344:
3336:and a right
3328:
3317:
3311:
3307:
3303:
3298:
3292:
3286:
3281:
3275:
3271:
3266:
3260:
3256:
3253:
3246:
3242:
3238:
3231:
3227:
3223:
3221:and labeled
3218:
3105:
3096:
3090:
3066:
3060:
3054:
3050:
3044:
3040:
3036:
3029:
3025:
3021:
3017:
3012:
3008:
3003:
2993:
2986:
2980:
2976:
2972:
2968:
2959:
2942:
2938:
2930:
2923:
2917:(the second
2913:
2909:
2902:
2891:
2887:
2881:
2877:
2871:which (when
2866:
2862:
2855:
2851:
2841:
2837:
2833:
2830:
2824:
2820:
2813:
2809:
2802:
2798:
2794:
2790:
2785:
2780:
2776:
2772:
2768:
2765:
2759:
2755:
2751:
2747:
2743:
2731:
2725:
2721:
2717:
2702:
2690:
2682:
2678:
2672:
2668:
2662:
2658:
2651:
2647:
2643:
2639:
2635:
2631:
2627:
2621:
2617:
2613:
2604:
2600:
2596:
2592:
2589:
2505:
2501:
2494:
2485:
2479:
2473:
2472:
2393:
2389:
2385:
2379:
2374:
2368:
2359:
2358:
2346:
2342:
2335:
2325:
2321:
2318:
2218:
2212:
2206:
2200:
2193:
2190:
2186:
2180:
2174:
2168:
2163:
2159:
2153:
2150:
2143:
2139:
2135:
2129:
2127:
2123:
2119:
2115:
2110:
2106:
2102:
2098:
2093:
2087:
2083:
2079:
2073:
2069:
2065:
2058:
2054:
2050:
2045:
2040:
2036:
2032:
2028:
2023:
2011:
2005:
1879:
1872:
1868:
1861:
1854:
1851:
1844:
1828:
1821:
1809:
1799:
1795:
1791:
1787:
1783:
1761:
1755:
1748:
1744:
1740:
1736:
1732:
1728:
1719:
1711:
1700:
1693:
1689:
1685:
1678:
1667:
1661:
1654:
1650:
1646:
1642:
1638:
1634:
1627:
1622:
1611:
1607:
1603:
1599:
1591:non-singular
1568:
1562:
1559:
1554:
1550:
1546:
1542:
1538:
1534:
1528:
1524:
1519:
1515:
1511:
1508:
1502:
1498:
1493:
1487:
1483:
1479:
1474:
1470:
1467:
1461:
1452:
1446:
1433:
1424:
1417:
1407:
1398:
1387:
1378:
1369:
1359:
1357:of the form
1354:
1351:left radical
1350:
1348:
1343:
1339:
1335:
1331:
1327:
1323:
1316:
1311:
1306:
1300:
1287:
1278:
1265:
1252:
1246:
1235:
1229:
1225:
1219:
1215:
1209:
1205:
1201:
1197:
1190:
1186:
1177:
1175:
1167:
1091:
1011:
1002:
993:
986:
975:
969:
965:
961:
954:
946:
942:
938:
931:
926:
920:
916:
912:
908:
904:
897:
889:
885:
881:
877:
873:
866:
858:
854:
847:
840:
834:
817:
814:
792:
789:
653:
649:
642:
634:
499:
493:
486:
480:
474:
467:
464:
456:
452:
445:
439:
432:
428:
423:
419:
415:
408:
403:
398:
394:
391:
383:
379:
372:
354:
351:
332:
322:
316:replaced by
303:
267:
260:
256:
252:
248:
245:
241:
237:
224:
220:
216:
212:
208:
204:
200:
196:
192:
188:
180:
176:
172:
168:
164:
161:
157:
144:
140:
136:
132:
128:
124:
120:
116:
112:
108:
95:
91:
87:
83:
76:
72:
63:
57:vector space
51:
47:
43:
40:bilinear map
35:
29:
4749:Heat kernel
4739:Hardy space
4646:Trace class
4560:Hahn–Banach
4522:Topological
3997:Harvey 1990
3985:Harvey 1990
3880:Polar space
3870:Linear form
3836:double dual
3350:dual module
3102:quaternions
2997:, and from
2806:that sends
2625:defined by
2474:Definition:
2360:Definition:
2094:Definition:
2024:Definition:
1772:Proposition
1680:alternating
1587:determinant
1468:Definition:
364:dimensional
314:linear maps
270:dot product
32:mathematics
4886:Categories
4872:PlanetMath
4682:C*-algebra
4497:Properties
4377:PlanetMath
4248:0292.10016
4218:Milnor, J.
4142:0288.15002
4066:, Springer
4053:0768.00003
4019:References
3961:Grove 1997
3318:split-case
3280:is called
3265:is called
2847:dual space
2044:is called
1462:definition
1180:unimodular
831:dual space
800:Properties
747:where the
4656:Unbounded
4651:Transpose
4609:Operators
4538:Separable
4533:Reflexive
4518:Algebraic
4504:Barrelled
4364:EMS Press
4310:, Dover,
4077:CRC Press
3907:Citations
3828:. Here,
3798:′ :
3711: : (
3645: : (
3560: : (
3162:∑
3158:−
3118:∑
2539:≥
2427:≤
2331:bijective
2294:∈
2253:∣
2235:⊥
2046:reflexive
1973:−
1947:−
1835:symmetric
1629:symmetric
1312:transpose
1304:). Given
1261:transpose
1150:∈
1074:∈
838:. Define
687:∑
637:congruent
562:∑
68:) over a
4858:Category
4670:Algebras
4552:Theorems
4509:Complete
4478:Schwartz
4424:glossary
4290:Springer
4257:(1995),
4186:. Also:
4114:(1974),
4062:(1970),
3848:See also
3806: :
3771: :
3763: :
3741: :
3699: :
3680: :
3672: :
3633: :
3614: :
3606: :
3529:and all
3501:for all
3362: :
3348:and its
3329:Given a
3236:, where
2971: :
2941:) ≃ Sym(
2771: :
2638: :
2630: :
2616: :
2595: :
2555:‖
2547:‖
2490:coercive
2486:elliptic
2455:‖
2447:‖
2442:‖
2434:‖
2345:) − dim(
2213:Suppose
2101: :
2078:for all
2063:implies
2031: :
1753:for all
1698:for all
1659:for all
1514: :
1496:implies
1492:for all
1363:are the
1139:for all
1063:for all
982:Currying
780:form an
503:, then:
99:that is
86: :
4661:Unitary
4641:Nuclear
4626:Compact
4621:Bounded
4616:Adjoint
4590:Min–max
4483:Sobolev
4468:Nuclear
4458:Hilbert
4453:Fréchet
4418: (
4366:, 2001
4064:Algebra
3341:-module
2845:is the
2707:of the
2703:By the
2375:bounded
2164:radical
2162:or the
1810:If the
1365:kernels
1259:is the
829:to its
663:, then
491:matrix
472:matrix
465:If the
402:matrix
312:, with
308:over a
306:modules
78:scalars
65:vectors
4636:Normal
4473:Orlicz
4463:Hölder
4443:Banach
4432:Spaces
4420:topics
4336:
4314:
4296:
4269:
4246:
4236:
4207:
4160:
4140:
4130:
4101:
4083:
4051:
4041:
3515:, all
3278:−1, 1)
3100:, and
2927:). If
2693:> 1
2497:> 0
2482:, ‖⋅‖)
2371:, ‖⋅‖)
2189:= 0 ⇔
2160:kernel
2003:where
358:be an
101:linear
4448:Besov
3924:(PDF)
3719:) ↦ ⟨
3653:) ↦ ⟨
3284:. If
3043:) ↦ 2
2912:) ≃ Λ
2729:. If
2685:) = 2
2681:char(
2679:When
2665:) ≠ 2
2661:char(
2659:When
2488:, or
2146:) = 0
2076:) = 0
2061:) = 0
1882:) ≠ 2
1878:char(
1847:) ≠ 2
1843:char(
1824:) = 2
1820:char(
1778:Proof
1739:) = −
1696:) = 0
1610:) = 2
1579:basis
1490:) = 0
1422:then
1208:) = 2
648:, …,
451:, …,
378:, …,
368:basis
323:When
70:field
55:on a
38:is a
4796:(or
4514:Dual
4334:ISBN
4312:ISBN
4294:ISBN
4267:ISBN
4234:ISBN
4205:ISBN
4158:ISBN
4128:ISBN
4099:ISBN
4081:ISBN
4039:ISBN
3793:and
3568:) ↦
3482:) =
3454:) +
3442:) =
3410:) +
3398:) =
3331:ring
3263:, 0)
3033:via
2937:(Sym
2929:char
2890:(Sym
2689:dim
2687:and
2341:dim(
2126:are
2096:Let
1645:) =
1595:unit
1581:for
1545:) =
1531:via
1431:and
1416:dim(
1394:rank
1376:and
1353:and
1349:The
1334:) =
1272:(if
968:(⋅,
964:) =
949:, ⋅)
941:) =
911:) =
880:) =
392:The
352:Let
310:ring
268:The
251:) =
232:and
215:) +
203:) =
171:) =
152:and
135:) +
123:) =
34:, a
4244:Zbl
4138:Zbl
4049:Zbl
3838:of
3810:↦ (
3775:↦ (
3661:),
3588:on
3379:if
3297:or
3289:= 4
3080:or
3001:to
2991:to
2933:≠ 2
2921:of
2900:of
2849:of
2818:to
2484:is
2373:is
2339:is
2329:is
2322:V/W
2196:= 0
2133:if
2122:in
2086:in
2048:if
1849:).
1814:of
1765:in
1726:if
1718:or
1704:in
1683:if
1671:in
1632:if
1573:is
1569:If
1527:on
1477:if
1473:is
1444:to
1396:of
1388:If
1367:of
1314:of
1298:in
1236:If
1223:to
1184:if
1170:= 0
1096:and
1094:= 0
1000:or
984:).
862:by
821:on
489:× 1
470:× 1
272:on
30:In
4888::
4422:–
4374:.
4362:,
4356:,
4332:,
4292:,
4288:,
4265:,
4242:,
4232:,
4224:,
4152:,
4136:,
4126:,
4118:,
4075:,
4047:,
4037:,
4029:,
3844:.
3826:))
3822:,
3814:↦
3802:→
3791:))
3787:,
3779:↦
3767:→
3749:→
3745:×
3733:.
3731:)⟩
3723:,
3715:,
3707:→
3703:×
3684:↦
3676:→
3649:,
3641:→
3637:×
3618:↦
3610:→
3598:.
3593:×
3564:,
3556:→
3552:×
3543:.
3538:∈
3534:,
3524:∈
3520:,
3510:∈
3506:,
3490:,
3484:αB
3480:xβ
3478:,
3476:αu
3462:,
3450:,
3438:+
3434:,
3418:,
3406:,
3394:,
3390:+
3370:→
3366:×
3320:.
3310:,
3245:=
3241:+
3230:,
3069:ij
3058:.
3053:→
3045:xy
3039:,
3028:→
3024:×
3015:.
2979:→
2975:×
2947:.
2935:,
2908:(Λ
2885:.
2880:⊗
2865:⊗
2854:⊗
2840:→
2836:⊗
2828:.
2812:,
2801:⊗
2797:→
2793:×
2779:→
2775:⊗
2758:,
2750:↦
2746:⊗
2724:→
2720:⊗
2656:.
2650:,
2642:↦
2634:→
2620:→
2603:→
2599:×
2509:,
2504:∈
2397:,
2392:∈
2388:,
2351:.
2324:→
2187:Ax
2142:,
2118:,
2109:→
2105:×
2082:,
2072:,
2057:,
2039:→
2035:×
1871:→
1867::
1860:,
1798:+
1794:,
1790:+
1769:;
1759:,
1747:,
1735:,
1692:,
1665:,
1653:,
1641:,
1612:xy
1606:,
1557:).
1553:)(
1541:,
1518:→
1501:=
1486:,
1346:).
1342:,
1330:,
1228:=
1218:=
1210:xy
1204:,
1189:→
919:,
907:)(
888:,
876:)(
857:→
853::
846:,
796:.
793:AS
462:.
427:,
414:=
411:ij
397:×
389:.
320:.
259:,
253:λB
244:,
223:,
211:,
199:+
195:,
179:,
173:λB
167:,
143:,
131:,
119:,
115:+
94:→
90:×
50:→
46:×
4878:.
4800:)
4524:)
4520:/
4516:(
4426:)
4408:e
4401:t
4394:v
4380:.
3963:.
3841:M
3831:M
3824:x
3820:u
3818:(
3816:B
3812:u
3808:x
3804:M
3800:M
3796:T
3789:x
3785:u
3783:(
3781:B
3777:x
3773:u
3769:M
3765:M
3761:S
3756:R
3751:R
3747:M
3743:M
3739:B
3729:x
3727:(
3725:T
3721:u
3717:x
3713:u
3709:R
3705:M
3701:M
3697:B
3692:)
3690:x
3688:(
3686:T
3682:x
3678:M
3674:M
3670:T
3665:⟩
3663:x
3659:u
3657:(
3655:S
3651:x
3647:u
3643:R
3639:M
3635:M
3631:B
3626:)
3624:u
3622:(
3620:S
3616:u
3612:M
3608:M
3604:S
3595:M
3591:M
3576:)
3574:x
3572:(
3570:u
3566:x
3562:u
3558:R
3554:M
3550:M
3540:R
3536:β
3532:α
3526:M
3522:y
3518:x
3512:M
3508:v
3504:u
3496:β
3494:)
3492:x
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