1121:
25:
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293:
43:
846:
673:
836:
1145:
963:
818:
794:
1155:
686:
775:
666:
643:
61:
1045:
412:. Thus these norms are uniformly bounded. Passing to a subsequence if necessary, it can therefore be assumed that
690:
635:
157:
39:
841:
525:
1124:
897:
831:
659:
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861:
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106:
can be used to construct an orthonormal basis in the
Bergman space and hence an explicit form of the
1101:
917:
229:
be a square integrable holomorphic function on Ω, i.e. an element of the
Bergman space A(Ω). Define
1150:
953:
851:
754:
531:
103:
1050:
826:
111:
1081:
1025:
989:
489:
788:
189:, it has a subsequence, convergent on compacta in Ω. Since the inverse functions converge to
176:. In fact Carathéodory's theorem implies that the inverse maps tend uniformly on compacta to
784:
1064:
87:
651:
8:
1030:
968:
682:
94:
by complex polynomials. It states that complex polynomials form a dense subspace of the
1055:
922:
448:
83:
1035:
639:
156:
onto Ω, normalised to fix a given point in Ω with positive derivative there. By the
613:
1040:
958:
927:
907:
892:
887:
882:
608:
719:
599:
Farrell, O. J. (1934), "On approximation to an analytic function by polynomials",
902:
856:
804:
799:
770:
509:
729:
366:{\displaystyle \displaystyle {\|g_{n}\|_{\Omega _{n}}^{2}=\|g\|_{\Omega }^{2}.}}
86:(1908–1979) in 1934, is a result concerning the approximation in mean square of
1091:
943:
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107:
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1086:
739:
709:
99:
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1005:
912:
714:
75:
948:
780:
434:. Since the evaluation maps are continuous linear functions on A(Ω),
512:) from Harvard University in 1930 and spent his career from 1931 at
141:. By the Riemann mapping theorem there is a conformal mapping
516:
with a leave of absence from
January 1949 to May 1949 at the
681:
82:, proved independently by O. J. Farrell (1899–1981) and
297:
296:
623:
Theory of functions of a complex variable. Vol. III
34:
may be too technical for most readers to understand
1011:Spectral theory of ordinary differential equations
365:
529:
128:be bounded Jordan domains decreasing to Ω, with Ω
1137:
638:, vol. 21, American Mathematical Society,
193:, it follows that the subsequence converges to
667:
421:has a weak limit in A(Ω). On the other hand,
122:Let Ω be the bounded Jordan domain and let Ω
620:
568:
468:(Ω) generated by complex polynomials. Hence
344:
337:
313:
299:
674:
660:
172:) converges uniformly on compacta in Ω to
612:
532:"A History of the Mathematics Department"
62:Learn how and when to remove this message
46:, without removing the technical details.
964:Group algebra of a locally compact group
508:Orin J. Farrell received his PhD (under
98:of a domain bounded by a simple closed
598:
1138:
629:
582:
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655:
44:make it understandable to non-experts
18:
213:As a consequence the derivative of
110:, which in turn yields an explicit
13:
348:
318:
222:tends to 1 uniformly on compacta.
14:
1167:
1120:
1119:
1046:Topological quantum field theory
23:
1146:Theorems in functional analysis
636:Graduate Studies in Mathematics
614:10.1090/s0002-9904-1934-06002-6
430:tends uniformly on compacta to
576:
553:
502:
1:
842:Uniform boundedness principle
592:
526:Mathematics Genealogy Project
90:on a bounded open set in the
1156:Theorems in complex analysis
621:Markushevich, A. I. (1967),
547:Institute for Advanced Study
518:Institute for Advanced Study
472:lies in the weak closure of
460:lies in the closed subspace
134:containing the closure of Ω
80:Farrell–Markushevich theorem
7:
632:A course in operator theory
483:
158:Carathéodory kernel theorem
10:
1172:
985:Invariant subspace problem
530:Bick, Theodore A. (1993).
1115:
1074:
998:
977:
936:
875:
817:
763:
705:
698:
287:). By change of variable
180:. Given a subsequence of
954:Spectrum of a C*-algebra
630:Conway, John B. (2000),
495:
447:. On the other hand, by
117:
112:Riemann mapping function
1051:Noncommutative geometry
394:to Ω. Then the norm of
1107:Tomita–Takesaki theory
1082:Approximation property
1026:Calculus of variations
601:Bull. Amer. Math. Soc.
385:be the restriction of
367:
1102:Banach–Mazur distance
1065:Generalized functions
438:is the weak limit of
403:is less than that of
368:
88:holomorphic functions
847:Kakutani fixed-point
832:Riesz representation
294:
197:on compacta. Hence
104:Gram–Schmidt process
16:Mathematical theorem
1031:Functional calculus
990:Mahler's conjecture
969:Von Neumann algebra
683:Functional analysis
490:Mergelyan's theorem
357:
333:
1056:Riemann hypothesis
755:Topological vector
585:, pp. 151–152
565:, pp. 150–151
549:. 9 December 2019.
363:
362:
343:
312:
210:on compacta in Ω.
84:A. I. Markushevich
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1036:Integral operator
813:
812:
569:Markushevich 1967
543:"Orin J. Farrell"
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71:
64:
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1123:
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1041:Jones polynomial
959:Operator algebra
703:
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571:, pp. 31–35
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1151:Operator theory
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1075:Advanced topics
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994:
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898:Hilbert–Schmidt
871:
862:Gelfand–Naimark
809:
759:
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680:
646:
625:, Prentice–Hall
607:(12): 908–914,
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522:Orin J. Farrell
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449:Runge's theorem
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40:help improve it
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1092:Choquet theory
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944:Banach algebra
940:
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867:Banach–Alaoglu
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823:
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795:Locally convex
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108:Bergman kernel
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1103:
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1097:Weak topology
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1021:Index theorem
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1007:
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988:
986:
983:
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978:Open problems
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845:
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835:
833:
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828:
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822:
820:
816:
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801:
798:
796:
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790:
786:
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772:
769:
768:
766:
762:
756:
753:
751:
748:
746:
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741:
738:
736:
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731:
728:
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723:
721:
718:
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688:
684:
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645:0-8218-2065-6
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561:
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548:
544:
537:
536:Union College
533:
527:
523:
519:
515:
514:Union College
511:
505:
501:
491:
488:
487:
481:
479:
475:
471:
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463:
458:
454:
450:
445:
441:
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433:
428:
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406:
401:
397:
392:
388:
383:
379:
358:
353:
340:
334:
329:
322:
307:
303:
290:
289:
288:
286:
281:
277:
273:
268:
264:
260:
256:
251:
247:
242:
236:
232:
228:
223:
220:
216:
211:
209:
206:converges to
204:
200:
196:
192:
187:
183:
179:
175:
171:
166:
162:
159:
154:
148:
144:
138:
132:
126:
115:
113:
109:
105:
101:
97:
96:Bergman space
93:
92:complex plane
89:
85:
81:
77:
66:
63:
55:
52:February 2024
45:
41:
35:
32:This article
30:
21:
20:
1087:Balanced set
1061:Distribution
999:Applications
852:Krein–Milman
837:Closed graph
631:
622:
604:
600:
578:
555:
546:
535:
504:
477:
473:
469:
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146:
142:
136:
130:
124:
121:
100:Jordan curve
79:
73:
58:
49:
33:
1016:Heat kernel
1006:Hardy space
913:Trace class
827:Hahn–Banach
789:Topological
583:Conway 2000
563:Conway 2000
510:J. L. Walsh
476:, which is
76:mathematics
1140:Categories
949:C*-algebra
764:Properties
593:References
923:Unbounded
918:Transpose
876:Operators
805:Separable
800:Reflexive
785:Algebraic
771:Barrelled
349:Ω
345:‖
338:‖
319:Ω
314:‖
300:‖
1125:Category
937:Algebras
819:Theorems
776:Complete
745:Schwartz
691:glossary
484:See also
480:itself.
928:Unitary
908:Nuclear
893:Compact
888:Bounded
883:Adjoint
857:Min–max
750:Sobolev
735:Nuclear
725:Hilbert
720:Fréchet
685: (
524:at the
38:Please
903:Normal
740:Orlicz
730:Hölder
710:Banach
699:Spaces
687:topics
642:
520:. See
102:. The
78:, the
715:Besov
559:See:
496:Notes
118:Proof
1063:(or
781:Dual
640:ISBN
376:Let
257:) =
238:on Ω
225:Let
150:of Ω
609:doi
464:of
451:,
244:by
139:+ 1
74:In
42:to
1142::
689:–
634:,
605:40
603:,
545:.
540:;
534:.
528:;
283:'(
274:))
1067:)
791:)
787:/
783:(
693:)
675:e
668:t
661:v
611::
538:.
478:K
474:K
470:g
466:A
462:K
457:n
453:h
444:n
440:h
436:g
432:g
427:n
423:h
418:n
414:h
409:n
405:g
400:n
396:h
391:n
387:g
382:n
378:h
359:.
354:2
341:g
335:=
330:2
323:n
308:n
304:g
285:z
280:n
276:f
272:z
270:(
267:n
263:f
261:(
259:g
255:z
253:(
250:n
246:g
241:n
235:n
231:g
227:g
219:n
215:f
208:z
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195:z
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165:n
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59:(
54:)
50:(
36:.
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