1162:
540:
The noncommutative analog of the Banach-Stone theorem is the folklore theorem that two unital C*-algebras are isomorphic if and only if they are completely isometric (i.e., isometric at all matrix levels). Mere isometry is not enough, as shown by the existence of a C*-algebra that is not isomorphic
402:
282:
2544:
2327:
653:
2417:
2454:
523:
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419:
is due to Banach, while the extension to compact
Hausdorff spaces is due to Stone. In fact, they both prove a slight generalization—they do not assume that
1371:
2651:
1705:
1051:
2350:
478:
The Banach–Stone theorem has some generalizations for vector-valued continuous functions on compact, Hausdorff topological spaces. For example, if
293:
1827:
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1359:
2646:
714:
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487:
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1004:
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835:
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1837:
1920:
89:), the set of algebra homomorphisms into the scalar field, equipped with the weak*-topology inherited from the dual space
1387:
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1344:
816:
707:
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2156:
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2009:
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2004:
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1101:
1025:
907:
2432:
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1999:
1982:
1800:
1645:
1314:
1142:
958:
666:. Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej.
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47:
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1303:
1297:
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1066:
1030:
428:
140:
2097:
2641:
2636:
2111:
2059:
2016:
1940:
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1630:
1292:
1259:
1232:
829:
1930:
825:
438:
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1785:
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1491:
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1105:
641:
692:
671:
8:
2422:
2223:
2180:
1994:
1717:
1447:
1254:
1071:
1009:
723:
187:
25:
2368:
97:)*. The Banach-Stone theorem avoids reference to multiplicative structure by recovering
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2190:
2160:
1977:
1935:
1542:
1452:
1397:
1244:
1096:
963:
2498:
2468:
2429:
2337:
1810:
1591:
1534:
1514:
1076:
629:
29:
1967:
1962:
1950:
1862:
1847:
1710:
1650:
1625:
1556:
1546:
1409:
1081:
999:
968:
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667:
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1354:
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1987:
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190:
121:
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541:
to its opposite algebra (which trivially has the same Banach space structure).
40:
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1249:
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1061:
790:
775:
765:
649:
633:
416:
397:{\displaystyle (Tf)(y)=g(y)f(\varphi (y)){\mbox{ for all }}y\in Y,f\in C(X).}
194:
148:
118:
36:
33:
2074:
2069:
1529:
1519:
1392:
1382:
1227:
1207:
1127:
780:
750:
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136:
533:
from the extreme points of the duals of some other spaces of functions on
2363:
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2185:
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2150:
2124:
2089:
1640:
1603:
1276:
1056:
1046:
953:
755:
17:
2119:
2100: ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (H
2084:
1925:
1673:
1435:
989:
821:
1176:
576:. Warszawa: Instytut Matematyczny Polskiej Akademii Nauk. p. 170.
2195:
1440:
1404:
606:
589:
2461:
2345:
2271:
2231:
2134:
1957:
424:
2266:
1349:
590:"Applications of the Theory of Boolean Rings to General Topology"
620:
Araujo, Jesús (2006). "The noncompact Banach–Stone theorem".
722:
46:
In brief, the Banach–Stone theorem allows one to recover a
529:
A similar technique has also been used to recover a space
65:. If one is allowed to invoke the algebra structure of
355:
259:
2586:
2512:
2474:
2435:
2383:
2286:
2237:
441:
296:
229:
61:) of continuous real- or complex-valued functions on
2614:
2538:
2487:
2448:
2411:
2321:
2255:
1052:Spectral theory of ordinary differential equations
462:
396:
276:
594:Transactions of the American Mathematical Society
2669:
277:{\displaystyle |g(y)|=1{\mbox{ for all }}y\in Y}
2539:{\displaystyle S\left(\mathbb {R} ^{n}\right)}
2647:Mathematical formulation of quantum mechanics
1192:
708:
53:from the Banach space structure of the space
553: – Normed vector space that is complete
101:from the extreme points of the unit ball of
498:are compact, then every linear isometry of
427:in the sense of metric spaces, and use the
1199:
1185:
715:
701:
2522:
2322:{\displaystyle B_{p,q}^{s}(\mathbb {R} )}
2312:
605:
1005:Group algebra of a locally compact group
2412:{\displaystyle L^{\lambda ,p}(\Omega )}
1206:
24:is a classical result in the theory of
2670:
2652:Ordinary Differential Equations (ODEs)
1766:Banach–Steinhaus (Uniform boundedness)
648:
619:
571:
139:of continuous real- or complex-valued
1180:
696:
587:
13:
2480:
2441:
2403:
2247:
473:
14:
2699:
2144:Subsets / set operations
1921:Differentiation in Fréchet spaces
73:) this is easy – we can identify
1161:
1160:
1087:Topological quantum field theory
655:Théorie des Opérations Linéaires
574:Théorie des opérations linéaires
2688:Theorems in functional analysis
2449:{\displaystyle \ell ^{\infty }}
158:Given compact Hausdorff spaces
2678:Theory of continuous functions
2609:
2590:
2406:
2400:
2316:
2308:
2250:
2244:
1838:Lomonosov's invariant subspace
1761:Banach–Schauder (open mapping)
580:
564:
457:
451:
423:is linear, only that it is an
388:
382:
351:
348:
342:
336:
330:
324:
315:
309:
306:
297:
248:
244:
238:
231:
1:
883:Uniform boundedness principle
557:
1723:Singular value decomposition
112:
7:
2488:{\displaystyle L^{\infty }}
2256:{\displaystyle ba(\Sigma )}
2125:Radially convex/Star-shaped
660:Theory of Linear Operations
544:
10:
2704:
2615:{\displaystyle W(X,L^{p})}
1026:Invariant subspace problem
622:Journal of Operator Theory
2629:
2214:
2161:Algebraic interior (core)
2143:
2052:
1886:
1776:Cauchy–Schwarz inequality
1731:
1659:
1505:
1419:Function space Topologies
1418:
1332:
1215:
1156:
1115:
1039:
1018:
977:
916:
858:
804:
746:
739:
995:Spectrum of a C*-algebra
588:Stone, Marshall (1937).
1092:Noncommutative geometry
572:Banach, Stefan (1932).
524:strong Banach–Stone map
48:compact Hausdorff space
2616:
2540:
2489:
2450:
2413:
2323:
2257:
1426:Banach–Mazur compactum
1216:Types of Banach spaces
1148:Tomita–Takesaki theory
1123:Approximation property
1067:Calculus of variations
470:is a linear isometry.
464:
463:{\displaystyle T-T(0)}
398:
278:
193:. Then there exists a
2642:Finite element method
2637:Differential operator
2617:
2541:
2490:
2451:
2414:
2324:
2258:
2098:Convex series related
1894:Abstract Wiener space
1821:hyperplane separation
1376:Minkowski functionals
1260:Polarization identity
1143:Banach–Mazur distance
1106:Generalized functions
465:
399:
279:
2584:
2510:
2472:
2433:
2381:
2284:
2235:
2224:Absolute continuity
1878:Schauder fixed-point
1868:Riesz representation
1828:Kakutani fixed-point
1796:Freudenthal spectral
1282:L-semi-inner product
888:Kakutani fixed-point
873:Riesz representation
439:
294:
227:
147:, equipped with the
26:continuous functions
22:Banach–Stone theorem
2307:
2045:measurable function
1995:Functional calculus
1858:Parseval's identity
1771:Bessel's inequality
1718:Polar decomposition
1497:Uniform convergence
1255:Inner product space
1072:Functional calculus
1031:Mahler's conjecture
1010:Von Neumann algebra
724:Functional analysis
357: for all
261: for all
2657:Validated numerics
2612:
2568:Sobolev inequality
2536:
2485:
2446:
2409:
2338:Bounded variation
2319:
2287:
2272:Banach coordinate
2253:
2191:Minkowski addition
1853:M. Riesz extension
1333:Banach spaces are:
1097:Riemann hypothesis
796:Topological vector
460:
435:is affine, and so
429:Mazur–Ulam theorem
394:
359:
274:
263:
32:, named after the
30:topological spaces
2665:
2664:
2377:Morrey–Campanato
2359:compact Hausdorff
2206:Relative interior
2060:Absolutely convex
2027:Projection-valued
1636:Strictly singular
1562:on Hilbert spaces
1323:of Hilbert spaces
1174:
1173:
1077:Integral operator
854:
853:
358:
262:
2695:
2621:
2619:
2618:
2613:
2608:
2607:
2575:Triebel–Lizorkin
2545:
2543:
2542:
2537:
2535:
2531:
2530:
2525:
2494:
2492:
2491:
2486:
2484:
2483:
2455:
2453:
2452:
2447:
2445:
2444:
2418:
2416:
2415:
2410:
2399:
2398:
2328:
2326:
2325:
2320:
2315:
2306:
2301:
2262:
2260:
2259:
2254:
2115:
2093:
2075:Balanced/Circled
1873:Robinson-Ursescu
1791:Eberlein–Šmulian
1711:Spectral theorem
1507:Linear operators
1304:Uniformly smooth
1201:
1194:
1187:
1178:
1177:
1164:
1163:
1082:Jones polynomial
1000:Operator algebra
744:
743:
717:
710:
703:
694:
693:
689:
687:
686:
680:
674:. Archived from
665:
645:
612:
611:
609:
584:
578:
577:
568:
469:
467:
466:
461:
403:
401:
400:
395:
360:
356:
283:
281:
280:
275:
264:
260:
251:
234:
2703:
2702:
2698:
2697:
2696:
2694:
2693:
2692:
2683:Operator theory
2668:
2667:
2666:
2661:
2625:
2603:
2599:
2585:
2582:
2581:
2580:Wiener amalgam
2550:Segal–Bargmann
2526:
2521:
2520:
2516:
2511:
2508:
2507:
2479:
2475:
2473:
2470:
2469:
2440:
2436:
2434:
2431:
2430:
2388:
2384:
2382:
2379:
2378:
2333:Birnbaum–Orlicz
2311:
2302:
2291:
2285:
2282:
2281:
2236:
2233:
2232:
2210:
2166:Bounding points
2139:
2113:
2091:
2048:
1899:Banach manifold
1882:
1806:Gelfand–Naimark
1727:
1701:Spectral theory
1669:Banach algebras
1661:Operator theory
1655:
1616:Pseudo-monotone
1599:Hilbert–Schmidt
1579:Densely defined
1501:
1414:
1328:
1211:
1205:
1175:
1170:
1152:
1116:Advanced topics
1111:
1035:
1014:
973:
939:Hilbert–Schmidt
912:
903:Gelfand–Naimark
850:
800:
735:
721:
684:
682:
678:
663:
616:
615:
607:10.2307/1989788
585:
581:
569:
565:
560:
547:
476:
474:Generalizations
440:
437:
436:
407:The case where
354:
295:
292:
291:
258:
247:
230:
228:
225:
224:
208:and a function
191:linear isometry
154:
122:Hausdorff space
115:
12:
11:
5:
2701:
2691:
2690:
2685:
2680:
2663:
2662:
2660:
2659:
2654:
2649:
2644:
2639:
2633:
2631:
2627:
2626:
2624:
2623:
2611:
2606:
2602:
2598:
2595:
2592:
2589:
2577:
2572:
2571:
2570:
2560:
2558:Sequence space
2555:
2547:
2534:
2529:
2524:
2519:
2515:
2503:
2502:
2501:
2496:
2482:
2478:
2459:
2458:
2457:
2443:
2439:
2420:
2408:
2405:
2402:
2397:
2394:
2391:
2387:
2374:
2366:
2361:
2348:
2343:
2335:
2330:
2318:
2314:
2310:
2305:
2300:
2297:
2294:
2290:
2277:
2269:
2264:
2252:
2249:
2246:
2243:
2240:
2229:
2220:
2218:
2212:
2211:
2209:
2208:
2198:
2193:
2188:
2183:
2178:
2173:
2168:
2163:
2153:
2147:
2145:
2141:
2140:
2138:
2137:
2132:
2127:
2122:
2117:
2109:
2095:
2087:
2082:
2077:
2072:
2067:
2062:
2056:
2054:
2050:
2049:
2047:
2046:
2036:
2035:
2034:
2029:
2024:
2014:
2013:
2012:
2007:
2002:
1992:
1991:
1990:
1985:
1980:
1975:
1973:Gelfand–Pettis
1970:
1965:
1955:
1954:
1953:
1948:
1943:
1938:
1933:
1923:
1918:
1913:
1908:
1907:
1906:
1896:
1890:
1888:
1884:
1883:
1881:
1880:
1875:
1870:
1865:
1860:
1855:
1850:
1845:
1840:
1835:
1830:
1825:
1824:
1823:
1813:
1808:
1803:
1798:
1793:
1788:
1783:
1778:
1773:
1768:
1763:
1758:
1753:
1748:
1746:Banach–Alaoglu
1743:
1741:Anderson–Kadec
1737:
1735:
1729:
1728:
1726:
1725:
1720:
1715:
1714:
1713:
1708:
1698:
1697:
1696:
1691:
1681:
1679:Operator space
1676:
1671:
1665:
1663:
1657:
1656:
1654:
1653:
1648:
1643:
1638:
1633:
1628:
1623:
1618:
1613:
1612:
1611:
1601:
1596:
1595:
1594:
1589:
1581:
1576:
1566:
1565:
1564:
1554:
1549:
1539:
1538:
1537:
1532:
1527:
1517:
1511:
1509:
1503:
1502:
1500:
1499:
1494:
1489:
1488:
1487:
1482:
1472:
1471:
1470:
1465:
1455:
1450:
1445:
1444:
1443:
1433:
1428:
1422:
1420:
1416:
1415:
1413:
1412:
1407:
1402:
1401:
1400:
1390:
1385:
1380:
1379:
1378:
1367:Locally convex
1364:
1363:
1362:
1352:
1347:
1342:
1336:
1334:
1330:
1329:
1327:
1326:
1319:Tensor product
1312:
1306:
1301:
1295:
1290:
1284:
1279:
1274:
1264:
1263:
1262:
1257:
1247:
1242:
1240:Banach lattice
1237:
1236:
1235:
1225:
1219:
1217:
1213:
1212:
1204:
1203:
1196:
1189:
1181:
1172:
1171:
1169:
1168:
1157:
1154:
1153:
1151:
1150:
1145:
1140:
1135:
1133:Choquet theory
1130:
1125:
1119:
1117:
1113:
1112:
1110:
1109:
1099:
1094:
1089:
1084:
1079:
1074:
1069:
1064:
1059:
1054:
1049:
1043:
1041:
1037:
1036:
1034:
1033:
1028:
1022:
1020:
1016:
1015:
1013:
1012:
1007:
1002:
997:
992:
987:
985:Banach algebra
981:
979:
975:
974:
972:
971:
966:
961:
956:
951:
946:
941:
936:
931:
926:
920:
918:
914:
913:
911:
910:
908:Banach–Alaoglu
905:
900:
895:
890:
885:
880:
875:
870:
864:
862:
856:
855:
852:
851:
849:
848:
843:
838:
836:Locally convex
833:
819:
814:
808:
806:
802:
801:
799:
798:
793:
788:
783:
778:
773:
768:
763:
758:
753:
747:
741:
737:
736:
720:
719:
712:
705:
697:
691:
690:
650:Banach, Stefan
646:
628:(2): 285–294.
614:
613:
600:(3): 375–481.
586:Theorem 83 of
579:
570:Théorème 3 of
562:
561:
559:
556:
555:
554:
546:
543:
475:
472:
459:
456:
453:
450:
447:
444:
405:
404:
393:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
353:
350:
347:
344:
341:
338:
335:
332:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
299:
285:
284:
273:
270:
267:
257:
254:
250:
246:
243:
240:
237:
233:
178:) →
152:
114:
111:
41:Marshall Stone
34:mathematicians
9:
6:
4:
3:
2:
2700:
2689:
2686:
2684:
2681:
2679:
2676:
2675:
2673:
2658:
2655:
2653:
2650:
2648:
2645:
2643:
2640:
2638:
2635:
2634:
2632:
2628:
2622:
2604:
2600:
2596:
2593:
2587:
2578:
2576:
2573:
2569:
2566:
2565:
2564:
2561:
2559:
2556:
2554:
2553:
2548:
2546:
2532:
2527:
2517:
2513:
2504:
2500:
2497:
2495:
2476:
2467:
2466:
2465:
2464:
2460:
2456:
2437:
2428:
2427:
2426:
2425:
2421:
2419:
2395:
2392:
2389:
2385:
2375:
2373:
2372:
2367:
2365:
2362:
2360:
2358:
2354:
2349:
2347:
2344:
2342:
2341:
2336:
2334:
2331:
2329:
2303:
2298:
2295:
2292:
2288:
2278:
2276:
2275:
2270:
2268:
2265:
2263:
2241:
2238:
2230:
2228:
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2207:
2203:
2199:
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2189:
2187:
2184:
2182:
2179:
2177:
2176:Extreme point
2174:
2172:
2169:
2167:
2164:
2162:
2158:
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2149:
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2055:
2053:Types of sets
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2044:
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2033:
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2028:
2025:
2023:
2020:
2019:
2018:
2015:
2011:
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1932:
1929:
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1924:
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1917:
1916:Convex series
1914:
1912:
1911:Bochner space
1909:
1905:
1902:
1901:
1900:
1897:
1895:
1892:
1891:
1889:
1885:
1879:
1876:
1874:
1871:
1869:
1866:
1864:
1863:Riesz's lemma
1861:
1859:
1856:
1854:
1851:
1849:
1848:Mazur's lemma
1846:
1844:
1841:
1839:
1836:
1834:
1831:
1829:
1826:
1822:
1819:
1818:
1817:
1814:
1812:
1809:
1807:
1804:
1802:
1801:Gelfand–Mazur
1799:
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1787:
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1779:
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1381:
1377:
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1316:
1313:
1311:
1307:
1305:
1302:
1300:) convex
1299:
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1269:
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1256:
1253:
1252:
1251:
1248:
1246:
1245:Grothendieck
1243:
1241:
1238:
1234:
1231:
1230:
1229:
1226:
1224:
1221:
1220:
1218:
1214:
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1202:
1197:
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1190:
1188:
1183:
1182:
1179:
1167:
1159:
1158:
1155:
1149:
1146:
1144:
1141:
1139:
1138:Weak topology
1136:
1134:
1131:
1129:
1126:
1124:
1121:
1120:
1118:
1114:
1107:
1103:
1100:
1098:
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1090:
1088:
1085:
1083:
1080:
1078:
1075:
1073:
1070:
1068:
1065:
1063:
1062:Index theorem
1060:
1058:
1055:
1053:
1050:
1048:
1045:
1044:
1042:
1038:
1032:
1029:
1027:
1024:
1023:
1021:
1019:Open problems
1017:
1011:
1008:
1006:
1003:
1001:
998:
996:
993:
991:
988:
986:
983:
982:
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947:
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869:
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842:
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834:
831:
827:
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797:
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792:
789:
787:
784:
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779:
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769:
767:
764:
762:
759:
757:
754:
752:
749:
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745:
742:
738:
733:
729:
725:
718:
713:
711:
706:
704:
699:
698:
695:
681:on 2014-01-11
677:
673:
669:
661:
657:
656:
651:
647:
643:
639:
635:
631:
627:
623:
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599:
595:
591:
583:
575:
567:
563:
552:
549:
548:
542:
538:
536:
532:
527:
525:
521:
517:
513:
509:
505:
501:
497:
493:
489:
486:with trivial
485:
481:
471:
454:
448:
445:
442:
434:
431:to show that
430:
426:
422:
418:
417:metric spaces
414:
410:
391:
385:
379:
376:
373:
370:
367:
364:
361:
345:
339:
333:
327:
321:
318:
312:
303:
300:
290:
289:
288:
271:
268:
265:
255:
252:
241:
235:
223:
222:
221:
219:
215:
212: ∈
211:
207:
204: →
203:
200: :
199:
196:
195:homeomorphism
192:
189:
185:
181:
177:
173:
170: :
169:
165:
161:
156:
150:
149:supremum norm
146:
142:
138:
135:) denote the
134:
130:
126:
123:
120:
110:
108:
104:
100:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
49:
44:
42:
38:
37:Stefan Banach
35:
31:
27:
23:
19:
2630:Applications
2551:
2462:
2423:
2370:
2356:
2352:
2339:
2273:
2225:
2112:Linear cone
2105:
2101:
2090:Convex cone
1983:Paley–Wiener
1843:Mackey–Arens
1833:Krein–Milman
1786:Closed range
1781:Closed graph
1751:Banach–Mazur
1631:Self-adjoint
1535:sesquilinear
1268:Polynomially
1208:Banach space
1128:Balanced set
1102:Distribution
1040:Applications
893:Krein–Milman
878:Closed graph
683:. Retrieved
676:the original
659:
654:
625:
621:
597:
593:
582:
573:
566:
551:Banach space
539:
534:
530:
528:
519:
515:
511:
507:
503:
499:
495:
491:
484:Banach space
479:
477:
432:
420:
415:are compact
412:
408:
406:
286:
217:
213:
209:
205:
201:
197:
183:
179:
175:
171:
167:
163:
159:
157:
144:
137:Banach space
132:
128:
124:
116:
106:
102:
98:
94:
90:
86:
82:
74:
70:
66:
62:
58:
54:
50:
45:
21:
15:
2351:Continuous
2186:Linear span
2171:Convex hull
2151:Affine hull
2010:holomorphic
1946:holomorphic
1926:Derivatives
1816:Hahn–Banach
1756:Banach–Saks
1674:C*-algebras
1641:Trace class
1604:Functionals
1492:Ultrastrong
1405:Quasinormed
1057:Heat kernel
1047:Hardy space
954:Trace class
868:Hahn–Banach
830:Topological
488:centralizer
18:mathematics
2672:Categories
2104:), and (Hw
2005:continuous
1941:functional
1689:C*-algebra
1574:Continuous
1436:Dual space
1410:Stereotype
1388:Metrizable
1315:Projective
990:C*-algebra
805:Properties
685:2020-07-11
672:0005.20901
558:References
287:such that
188:surjective
166:, suppose
2563:Sobolev W
2506:Schwartz
2481:∞
2442:∞
2438:ℓ
2404:Ω
2390:λ
2248:Σ
2130:Symmetric
2065:Absorbing
1978:regulated
1958:Integrals
1811:Goldstine
1646:Transpose
1583:Fredholm
1453:Ultraweak
1441:Dual norm
1372:Seminorms
1340:Barrelled
1310:Injective
1298:Uniformly
1272:Reflexive
964:Unbounded
959:Transpose
917:Operators
846:Separable
841:Reflexive
826:Algebraic
812:Barrelled
634:0379-4024
446:−
377:∈
365:∈
340:φ
269:∈
141:functions
113:Statement
77:with the
2499:weighted
2369:Hilbert
2346:Bs space
2216:Examples
2181:Interior
2157:Relative
2135:Zonotope
2114:(subset)
2092:(subset)
2043:Strongly
2022:Lebesgue
2017:Measures
1887:Analysis
1733:Theorems
1684:Spectrum
1609:positive
1592:operator
1530:operator
1520:Bilinear
1485:operator
1468:operator
1448:Operator
1345:Complete
1293:Strictly
1166:Category
978:Algebras
860:Theorems
817:Complete
786:Schwartz
732:glossary
652:(1932).
545:See also
425:isometry
79:spectrum
2364:Hardy H
2267:c space
2204:)
2159:)
2080:Bounded
1968:Dunford
1963:Bochner
1936:Gateaux
1931:Fréchet
1706:of ODEs
1651:Unitary
1626:Nuclear
1557:Compact
1547:Bounded
1515:Adjoint
1355:Fréchet
1350:F-space
1321: (
1317:)
1270:)
1250:Hilbert
1223:Asplund
969:Unitary
949:Nuclear
934:Compact
929:Bounded
924:Adjoint
898:Min–max
791:Sobolev
776:Nuclear
766:Hilbert
761:Fréchet
726: (
642:2242851
522:) is a
518:;
510:) onto
506:;
220:) with
186:) is a
119:compact
2280:Besov
2120:Radial
2085:Convex
2070:Affine
2039:Weakly
2032:Vector
1904:bundle
1694:radius
1621:Normal
1587:kernel
1552:Closed
1475:Strong
1393:Normed
1383:Mackey
1228:Banach
1210:topics
944:Normal
781:Orlicz
771:Hölder
751:Banach
740:Spaces
728:topics
670:
662:]
640:
632:
198:φ
127:, let
117:For a
20:, the
2355:with
2202:Quasi
2196:Polar
2000:Borel
1951:quasi
1480:polar
1463:polar
1277:Riesz
756:Besov
679:(PDF)
664:(PDF)
658:[
482:is a
2353:C(K)
1988:weak
1525:form
1458:Weak
1431:Dual
1398:norm
1360:tame
1233:list
1104:(or
822:Dual
630:ISSN
494:and
490:and
411:and
162:and
109:)*.
39:and
1570:Dis
668:Zbl
602:doi
151:‖·‖
143:on
81:of
28:on
16:In
2674::
2340:BV
2274:BK
2226:AC
2108:))
2041:/
1543:Un
730:–
638:MR
636:.
626:55
624:.
598:41
596:.
592:.
537:.
526:.
155:.
43:.
2610:)
2605:p
2601:L
2597:,
2594:X
2591:(
2588:W
2552:F
2533:)
2528:n
2523:R
2518:(
2514:S
2477:L
2463:L
2424:ℓ
2407:)
2401:(
2396:p
2393:,
2386:L
2371:H
2357:K
2317:)
2313:R
2309:(
2304:s
2299:q
2296:,
2293:p
2289:B
2251:)
2245:(
2242:a
2239:b
2200:(
2155:(
2106:x
2102:x
1572:)
1568:(
1545:)
1541:(
1374:/
1325:)
1308:(
1288:B
1286:(
1266:(
1200:e
1193:t
1186:v
1108:)
832:)
828:/
824:(
734:)
716:e
709:t
702:v
688:.
644:.
610:.
604::
535:X
531:X
520:E
516:Y
514:(
512:C
508:E
504:X
502:(
500:C
496:Y
492:X
480:E
458:)
455:0
452:(
449:T
443:T
433:T
421:T
413:Y
409:X
392:.
389:)
386:X
383:(
380:C
374:f
371:,
368:Y
362:y
352:)
349:)
346:y
343:(
337:(
334:f
331:)
328:y
325:(
322:g
319:=
316:)
313:y
310:(
307:)
304:f
301:T
298:(
272:Y
266:y
256:1
253:=
249:|
245:)
242:y
239:(
236:g
232:|
218:Y
216:(
214:C
210:g
206:X
202:Y
184:Y
182:(
180:C
176:X
174:(
172:C
168:T
164:Y
160:X
153:∞
145:X
133:X
131:(
129:C
125:X
107:X
105:(
103:C
99:X
95:X
93:(
91:C
87:X
85:(
83:C
75:X
71:X
69:(
67:C
63:X
59:X
57:(
55:C
51:X
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