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Krein–Smulian theorem

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2807: 2726: 2750: 425: 2133: 1916: 2006: 2043: 2082: 1850: 456: 294: 2209: 347: 343: 476: 314: 267: 243: 212: 192: 168: 148: 128: 108: 84: 960: 2848: 2615: 2240: 1294: 1939: 2451: 1416: 948: 2278: 2235: 2791: 2441: 1349: 2882: 2568: 2423: 1321: 2399: 955: 787: 2841: 1668: 1610: 736: 706: 672: 609: 2887: 1426: 664: 1509: 976: 2877: 2291: 636: 2834: 2380: 2271: 1150: 933: 2098: 2650: 1745: 1068: 911: 2784: 2295: 1598: 1534: 1085: 1379: 2892: 1872: 1790: 1593: 1272: 1051: 2446: 1969: 1631: 1409: 1364: 1354: 698: 597: 2729: 2502: 2436: 2264: 1466: 1456: 1384: 1311: 1187: 856: 1461: 2466: 1804: 1794: 1394: 780: 2822: 2777: 2765: 2711: 2665: 2589: 2471: 2163: 1965: 1627: 1441: 1334: 1329: 1224: 1197: 1162: 1014: 907: 2706: 2522: 2021: 1921: 1615: 1588: 1571: 1389: 1234: 903: 663:. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: 2558: 2456: 2359: 2138: 1431: 1421: 1339: 1277: 1204: 1158: 1073: 898: 488: 2655: 2431: 2060: 1823: 1404: 1344: 434: 272: 2172: 420:{\displaystyle A\cap \left\{x^{\prime }\in X^{\prime }:\left\|x^{\prime }\right\|\leq r\right\}} 2872: 2867: 2686: 2630: 2594: 1446: 1359: 1140: 1056: 892: 886: 773: 1686: 2757: 2393: 2230: 2225: 1700: 1648: 1605: 1529: 1482: 1219: 881: 848: 821: 527: 319: 2389: 1519: 2669: 2168: 1374: 1369: 1080: 964: 870: 724: 556: 2256: 760: 8: 2814: 2635: 2573: 2287: 2011: 1812: 1769: 1583: 1306: 1036: 843: 601: 21: 1957: 2660: 2527: 2245: 2156: 1779: 1749: 1566: 1524: 1131: 1041: 986: 833: 690: 544: 461: 428: 299: 252: 228: 197: 177: 153: 133: 113: 93: 69: 2087: 2057: 2018: 2640: 1926: 1399: 1180: 1123: 1103: 742: 732: 712: 702: 678: 668: 658: 642: 632: 615: 605: 2645: 2563: 2532: 2512: 2497: 2492: 2487: 1556: 1551: 1539: 1451: 1436: 1299: 1239: 1214: 1145: 1135: 998: 701:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. 536: 2324: 943: 2507: 2461: 2409: 2404: 2375: 1576: 1561: 1487: 1289: 1282: 1249: 1209: 1175: 1167: 1095: 1063: 928: 860: 552: 522: 494: 45: 2334: 2818: 2761: 2696: 2548: 2349: 2146: 2094: 1754: 1620: 1267: 1257: 876: 828: 589: 2861: 2701: 2625: 2354: 2339: 2329: 2151: 1764: 1718: 1653: 1504: 1499: 1492: 1113: 1046: 1019: 838: 811: 746: 716: 646: 631:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. 525:(1940). "On regularly convex sets in the space conjugate to a Banach space". 171: 56:
Both of the following theorems are referred to as the Krein-Smulian Theorem.
37: 33: 682: 619: 2691: 2344: 2314: 1663: 1658: 1118: 1108: 981: 971: 816: 796: 654: 491: – On when a space equals the closed convex hull of its extreme points 246: 87: 2620: 2610: 2517: 2319: 1952: 1868: 1774: 1759: 1739: 1713: 1678: 1229: 1192: 865: 29: 17: 2553: 2385: 1708: 1689: ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (H 1673: 1514: 1262: 1024: 548: 518: 41: 2806: 765: 1784: 1029: 993: 540: 2050: 1934: 1860: 1820: 1723: 1546: 1855: 938: 2749: 2286: 568: 566: 563: 761:
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Topological Vector Spaces, Distributions and Kernels
2616:Spectral theory of ordinary differential equations 2203: 2127: 2076: 2037: 2000: 1910: 1844: 470: 450: 419: 337: 308: 288: 261: 237: 206: 186: 162: 142: 122: 102: 78: 626: 2859: 2128:{\displaystyle S\left(\mathbb {R} ^{n}\right)} 627:Narici, Lawrence; Beckenstein, Edward (2011). 28:can refer to two theorems relating the closed 2842: 2785: 2272: 2236:Mathematical formulation of quantum mechanics 781: 689: 269:a convex subset of the continuous dual space 517: 2849: 2835: 2792: 2778: 2279: 2265: 788: 774: 2111: 1911:{\displaystyle B_{p,q}^{s}(\mathbb {R} )} 1901: 600:. Vol. 96 (2nd ed.). New York: 2569:Group algebra of a locally compact group 2001:{\displaystyle L^{\lambda ,p}(\Omega )} 795: 2860: 2241:Ordinary Differential Equations (ODEs) 1355:Banach–Steinhaus (Uniform boundedness) 723: 588: 572: 2260: 769: 731:. Mineola, N.Y.: Dover Publications. 653: 2801: 2744: 665:McGraw-Hill Science/Engineering/Math 13: 2069: 2030: 1992: 1836: 754: 443: 397: 380: 367: 281: 174:). Then the closed convex hull of 14: 2904: 1733:Subsets / set operations 1510:Differentiation in Fréchet spaces 2805: 2748: 2725: 2724: 2651:Topological quantum field theory 2883:Theorems in functional analysis 2038:{\displaystyle \ell ^{\infty }} 594:A Course in Functional Analysis 582: 2198: 2179: 1995: 1989: 1905: 1897: 1839: 1833: 1427:Lomonosov's invariant subspace 1350:Banach–Schauder (open mapping) 511: 402: 389: 48:, who published them in 1940. 1: 2447:Uniform boundedness principle 598:Graduate Texts in Mathematics 504: 2821:. You can help Knowledge by 2764:. You can help Knowledge by 1312:Singular value decomposition 693:; Wolff, Manfred P. (1999). 51: 7: 2888:Mathematical analysis stubs 2077:{\displaystyle L^{\infty }} 1845:{\displaystyle ba(\Sigma )} 1714:Radially convex/Star-shaped 482: 451:{\displaystyle X^{\prime }} 289:{\displaystyle X^{\prime }} 110:a weakly compact subset of 10: 2909: 2800: 2743: 2590:Invariant subspace problem 2204:{\displaystyle W(X,L^{p})} 2878:Topological vector spaces 2720: 2679: 2603: 2582: 2541: 2480: 2422: 2368: 2310: 2303: 2218: 1803: 1750:Algebraic interior (core) 1732: 1641: 1475: 1365:Cauchy–Schwarz inequality 1320: 1248: 1094: 1008:Function space Topologies 1007: 921: 804: 695:Topological Vector Spaces 629:Topological Vector Spaces 497: – Mathematical term 2559:Spectrum of a C*-algebra 2656:Noncommutative geometry 338:{\displaystyle r>0,} 40:. They are named after 2817:-related article is a 2760:–related article is a 2712:Tomita–Takesaki theory 2687:Approximation property 2631:Calculus of variations 2205: 2129: 2078: 2039: 2002: 1912: 1846: 1015:Banach–Mazur compactum 805:Types of Banach spaces 472: 452: 421: 339: 310: 290: 263: 239: 208: 188: 164: 144: 124: 104: 80: 61:Krein-Smulian Theorem: 2893:Metric geometry stubs 2758:mathematical analysis 2707:Banach–Mazur distance 2670:Generalized functions 2231:Finite element method 2226:Differential operator 2206: 2130: 2079: 2040: 2003: 1913: 1847: 1687:Convex series related 1483:Abstract Wiener space 1410:hyperplane separation 965:Minkowski functionals 849:Polarization identity 528:Annals of Mathematics 473: 453: 422: 340: 311: 291: 264: 240: 220:Krein-Smulian Theorem 209: 189: 165: 145: 125: 105: 81: 26:Krein-Smulian theorem 2452:Kakutani fixed-point 2437:Riesz representation 2173: 2099: 2061: 2022: 1970: 1873: 1824: 1813:Absolute continuity 1467:Schauder fixed-point 1457:Riesz representation 1417:Kakutani fixed-point 1385:Freudenthal spectral 871:L-semi-inner product 489:Krein–Milman theorem 462: 435: 348: 320: 300: 273: 253: 229: 198: 178: 170:is endowed with the 154: 134: 114: 94: 70: 2815:hyperbolic geometry 2636:Functional calculus 2595:Mahler's conjecture 2574:Von Neumann algebra 2288:Functional analysis 1896: 1634:measurable function 1584:Functional calculus 1447:Parseval's identity 1360:Bessel's inequality 1307:Polar decomposition 1086:Uniform convergence 844:Inner product space 691:Schaefer, Helmut H. 660:Functional Analysis 575:, pp. 159–165. 223: —  214:is weakly compact. 64: —  22:functional analysis 2661:Riemann hypothesis 2360:Topological vector 2246:Validated numerics 2201: 2157:Sobolev inequality 2125: 2074: 2035: 1998: 1927:Bounded variation 1908: 1876: 1861:Banach coordinate 1842: 1780:Minkowski addition 1442:M. Riesz extension 922:Banach spaces are: 478:is weak-* closed. 468: 448: 417: 335: 306: 286: 259: 235: 221: 204: 184: 160: 140: 120: 100: 76: 62: 20:, particularly in 2830: 2829: 2773: 2772: 2738: 2737: 2641:Integral operator 2418: 2417: 2254: 2253: 1966:Morrey–Campanato 1948:compact Hausdorff 1795:Relative interior 1649:Absolutely convex 1616:Projection-valued 1225:Strictly singular 1151:on Hilbert spaces 912:of Hilbert spaces 738:978-0-486-45352-1 708:978-1-4612-7155-0 674:978-0-07-054236-5 611:978-0-387-97245-9 531:. Second Series. 471:{\displaystyle A} 309:{\displaystyle X} 262:{\displaystyle A} 238:{\displaystyle X} 219: 207:{\displaystyle X} 187:{\displaystyle K} 163:{\displaystyle X} 143:{\displaystyle K} 123:{\displaystyle X} 103:{\displaystyle K} 79:{\displaystyle X} 60: 2900: 2851: 2844: 2837: 2809: 2802: 2794: 2787: 2780: 2752: 2745: 2728: 2727: 2646:Jones polynomial 2564:Operator algebra 2308: 2307: 2281: 2274: 2267: 2258: 2257: 2210: 2208: 2207: 2202: 2197: 2196: 2164:Triebel–Lizorkin 2134: 2132: 2131: 2126: 2124: 2120: 2119: 2114: 2083: 2081: 2080: 2075: 2073: 2072: 2044: 2042: 2041: 2036: 2034: 2033: 2007: 2005: 2004: 1999: 1988: 1987: 1917: 1915: 1914: 1909: 1904: 1895: 1890: 1851: 1849: 1848: 1843: 1704: 1682: 1664:Balanced/Circled 1462:Robinson-Ursescu 1380:Eberlein–Šmulian 1300:Spectral theorem 1096:Linear operators 893:Uniformly smooth 790: 783: 776: 767: 766: 750: 725:Trèves, François 720: 686: 650: 623: 576: 570: 561: 560: 515: 500: 477: 475: 474: 469: 457: 455: 454: 449: 447: 446: 426: 424: 423: 418: 416: 412: 405: 401: 400: 384: 383: 371: 370: 344: 342: 341: 336: 315: 313: 312: 307: 295: 293: 292: 287: 285: 284: 268: 266: 265: 260: 244: 242: 241: 236: 224: 213: 211: 210: 205: 193: 191: 190: 185: 169: 167: 166: 161: 150:is compact when 149: 147: 146: 141: 129: 127: 126: 121: 109: 107: 106: 101: 85: 83: 82: 77: 65: 2908: 2907: 2903: 2902: 2901: 2899: 2898: 2897: 2858: 2857: 2856: 2855: 2799: 2798: 2741: 2739: 2734: 2716: 2680:Advanced topics 2675: 2599: 2578: 2537: 2503:Hilbert–Schmidt 2476: 2467:Gelfand–Naimark 2414: 2364: 2299: 2285: 2255: 2250: 2214: 2192: 2188: 2174: 2171: 2170: 2169:Wiener amalgam 2139:Segal–Bargmann 2115: 2110: 2109: 2105: 2100: 2097: 2096: 2068: 2064: 2062: 2059: 2058: 2029: 2025: 2023: 2020: 2019: 1977: 1973: 1971: 1968: 1967: 1922:Birnbaum–Orlicz 1900: 1891: 1880: 1874: 1871: 1870: 1825: 1822: 1821: 1799: 1755:Bounding points 1728: 1702: 1680: 1637: 1488:Banach manifold 1471: 1395:Gelfand–Naimark 1316: 1290:Spectral theory 1258:Banach algebras 1250:Operator theory 1244: 1205:Pseudo-monotone 1188:Hilbert–Schmidt 1168:Densely defined 1090: 1003: 917: 800: 794: 757: 755:Further reading 739: 709: 675: 639: 612: 602:Springer-Verlag 590:Conway, John B. 585: 580: 579: 571: 564: 541:10.2307/1968735 516: 512: 507: 498: 495:Weak-* topology 485: 480: 463: 460: 459: 442: 438: 436: 433: 432: 396: 392: 388: 379: 375: 366: 362: 361: 357: 349: 346: 345: 321: 318: 317: 301: 298: 297: 280: 276: 274: 271: 270: 254: 251: 250: 230: 227: 226: 222: 216: 199: 196: 195: 179: 176: 175: 155: 152: 151: 135: 132: 131: 115: 112: 111: 95: 92: 91: 71: 68: 67: 63: 54: 46:Vitold Shmulyan 12: 11: 5: 2906: 2896: 2895: 2890: 2885: 2880: 2875: 2870: 2854: 2853: 2846: 2839: 2831: 2828: 2827: 2810: 2797: 2796: 2789: 2782: 2774: 2771: 2770: 2753: 2736: 2735: 2733: 2732: 2721: 2718: 2717: 2715: 2714: 2709: 2704: 2699: 2697:Choquet theory 2694: 2689: 2683: 2681: 2677: 2676: 2674: 2673: 2663: 2658: 2653: 2648: 2643: 2638: 2633: 2628: 2623: 2618: 2613: 2607: 2605: 2601: 2600: 2598: 2597: 2592: 2586: 2584: 2580: 2579: 2577: 2576: 2571: 2566: 2561: 2556: 2551: 2549:Banach algebra 2545: 2543: 2539: 2538: 2536: 2535: 2530: 2525: 2520: 2515: 2510: 2505: 2500: 2495: 2490: 2484: 2482: 2478: 2477: 2475: 2474: 2472:Banach–Alaoglu 2469: 2464: 2459: 2454: 2449: 2444: 2439: 2434: 2428: 2426: 2420: 2419: 2416: 2415: 2413: 2412: 2407: 2402: 2400:Locally convex 2397: 2383: 2378: 2372: 2370: 2366: 2365: 2363: 2362: 2357: 2352: 2347: 2342: 2337: 2332: 2327: 2322: 2317: 2311: 2305: 2301: 2300: 2284: 2283: 2276: 2269: 2261: 2252: 2251: 2249: 2248: 2243: 2238: 2233: 2228: 2222: 2220: 2216: 2215: 2213: 2212: 2200: 2195: 2191: 2187: 2184: 2181: 2178: 2166: 2161: 2160: 2159: 2149: 2147:Sequence space 2144: 2136: 2123: 2118: 2113: 2108: 2104: 2092: 2091: 2090: 2085: 2071: 2067: 2048: 2047: 2046: 2032: 2028: 2009: 1997: 1994: 1991: 1986: 1983: 1980: 1976: 1963: 1955: 1950: 1937: 1932: 1924: 1919: 1907: 1903: 1899: 1894: 1889: 1886: 1883: 1879: 1866: 1858: 1853: 1841: 1838: 1835: 1832: 1829: 1818: 1809: 1807: 1801: 1800: 1798: 1797: 1787: 1782: 1777: 1772: 1767: 1762: 1757: 1752: 1742: 1736: 1734: 1730: 1729: 1727: 1726: 1721: 1716: 1711: 1706: 1698: 1684: 1676: 1671: 1666: 1661: 1656: 1651: 1645: 1643: 1639: 1638: 1636: 1635: 1625: 1624: 1623: 1618: 1613: 1603: 1602: 1601: 1596: 1591: 1581: 1580: 1579: 1574: 1569: 1564: 1562:Gelfand–Pettis 1559: 1554: 1544: 1543: 1542: 1537: 1532: 1527: 1522: 1512: 1507: 1502: 1497: 1496: 1495: 1485: 1479: 1477: 1473: 1472: 1470: 1469: 1464: 1459: 1454: 1449: 1444: 1439: 1434: 1429: 1424: 1419: 1414: 1413: 1412: 1402: 1397: 1392: 1387: 1382: 1377: 1372: 1367: 1362: 1357: 1352: 1347: 1342: 1337: 1335:Banach–Alaoglu 1332: 1330:Anderson–Kadec 1326: 1324: 1318: 1317: 1315: 1314: 1309: 1304: 1303: 1302: 1297: 1287: 1286: 1285: 1280: 1270: 1268:Operator space 1265: 1260: 1254: 1252: 1246: 1245: 1243: 1242: 1237: 1232: 1227: 1222: 1217: 1212: 1207: 1202: 1201: 1200: 1190: 1185: 1184: 1183: 1178: 1170: 1165: 1155: 1154: 1153: 1143: 1138: 1128: 1127: 1126: 1121: 1116: 1106: 1100: 1098: 1092: 1091: 1089: 1088: 1083: 1078: 1077: 1076: 1071: 1061: 1060: 1059: 1054: 1044: 1039: 1034: 1033: 1032: 1022: 1017: 1011: 1009: 1005: 1004: 1002: 1001: 996: 991: 990: 989: 979: 974: 969: 968: 967: 956:Locally convex 953: 952: 951: 941: 936: 931: 925: 923: 919: 918: 916: 915: 908:Tensor product 901: 895: 890: 884: 879: 873: 868: 863: 853: 852: 851: 846: 836: 831: 829:Banach lattice 826: 825: 824: 814: 808: 806: 802: 801: 793: 792: 785: 778: 770: 764: 763: 756: 753: 752: 751: 737: 721: 707: 687: 673: 651: 638:978-1584888666 637: 624: 610: 584: 581: 578: 577: 562: 535:(3): 556–583. 509: 508: 506: 503: 502: 501: 492: 484: 481: 467: 445: 441: 415: 411: 408: 404: 399: 395: 391: 387: 382: 378: 374: 369: 365: 360: 356: 353: 334: 331: 328: 325: 305: 283: 279: 258: 234: 217: 203: 183: 159: 139: 119: 99: 75: 58: 53: 50: 9: 6: 4: 3: 2: 2905: 2894: 2891: 2889: 2886: 2884: 2881: 2879: 2876: 2874: 2873:Normed spaces 2871: 2869: 2868:Banach spaces 2866: 2865: 2863: 2852: 2847: 2845: 2840: 2838: 2833: 2832: 2826: 2824: 2820: 2816: 2811: 2808: 2804: 2803: 2795: 2790: 2788: 2783: 2781: 2776: 2775: 2769: 2767: 2763: 2759: 2754: 2751: 2747: 2746: 2742: 2731: 2723: 2722: 2719: 2713: 2710: 2708: 2705: 2703: 2702:Weak topology 2700: 2698: 2695: 2693: 2690: 2688: 2685: 2684: 2682: 2678: 2671: 2667: 2664: 2662: 2659: 2657: 2654: 2652: 2649: 2647: 2644: 2642: 2639: 2637: 2634: 2632: 2629: 2627: 2626:Index theorem 2624: 2622: 2619: 2617: 2614: 2612: 2609: 2608: 2606: 2602: 2596: 2593: 2591: 2588: 2587: 2585: 2583:Open problems 2581: 2575: 2572: 2570: 2567: 2565: 2562: 2560: 2557: 2555: 2552: 2550: 2547: 2546: 2544: 2540: 2534: 2531: 2529: 2526: 2524: 2521: 2519: 2516: 2514: 2511: 2509: 2506: 2504: 2501: 2499: 2496: 2494: 2491: 2489: 2486: 2485: 2483: 2479: 2473: 2470: 2468: 2465: 2463: 2460: 2458: 2455: 2453: 2450: 2448: 2445: 2443: 2440: 2438: 2435: 2433: 2430: 2429: 2427: 2425: 2421: 2411: 2408: 2406: 2403: 2401: 2398: 2395: 2391: 2387: 2384: 2382: 2379: 2377: 2374: 2373: 2371: 2367: 2361: 2358: 2356: 2353: 2351: 2348: 2346: 2343: 2341: 2338: 2336: 2333: 2331: 2328: 2326: 2323: 2321: 2318: 2316: 2313: 2312: 2309: 2306: 2302: 2297: 2293: 2289: 2282: 2277: 2275: 2270: 2268: 2263: 2262: 2259: 2247: 2244: 2242: 2239: 2237: 2234: 2232: 2229: 2227: 2224: 2223: 2221: 2217: 2211: 2193: 2189: 2185: 2182: 2176: 2167: 2165: 2162: 2158: 2155: 2154: 2153: 2150: 2148: 2145: 2143: 2142: 2137: 2135: 2121: 2116: 2106: 2102: 2093: 2089: 2086: 2084: 2065: 2056: 2055: 2054: 2053: 2049: 2045: 2026: 2017: 2016: 2015: 2014: 2010: 2008: 1984: 1981: 1978: 1974: 1964: 1962: 1961: 1956: 1954: 1951: 1949: 1947: 1943: 1938: 1936: 1933: 1931: 1930: 1925: 1923: 1920: 1918: 1892: 1887: 1884: 1881: 1877: 1867: 1865: 1864: 1859: 1857: 1854: 1852: 1830: 1827: 1819: 1817: 1816: 1811: 1810: 1808: 1806: 1802: 1796: 1792: 1788: 1786: 1783: 1781: 1778: 1776: 1773: 1771: 1768: 1766: 1765:Extreme point 1763: 1761: 1758: 1756: 1753: 1751: 1747: 1743: 1741: 1738: 1737: 1735: 1731: 1725: 1722: 1720: 1717: 1715: 1712: 1710: 1707: 1705: 1699: 1696: 1692: 1688: 1685: 1683: 1677: 1675: 1672: 1670: 1667: 1665: 1662: 1660: 1657: 1655: 1652: 1650: 1647: 1646: 1644: 1642:Types of sets 1640: 1633: 1629: 1626: 1622: 1619: 1617: 1614: 1612: 1609: 1608: 1607: 1604: 1600: 1597: 1595: 1592: 1590: 1587: 1586: 1585: 1582: 1578: 1575: 1573: 1570: 1568: 1565: 1563: 1560: 1558: 1555: 1553: 1550: 1549: 1548: 1545: 1541: 1538: 1536: 1533: 1531: 1528: 1526: 1523: 1521: 1518: 1517: 1516: 1513: 1511: 1508: 1506: 1505:Convex series 1503: 1501: 1500:Bochner space 1498: 1494: 1491: 1490: 1489: 1486: 1484: 1481: 1480: 1478: 1474: 1468: 1465: 1463: 1460: 1458: 1455: 1453: 1452:Riesz's lemma 1450: 1448: 1445: 1443: 1440: 1438: 1437:Mazur's lemma 1435: 1433: 1430: 1428: 1425: 1423: 1420: 1418: 1415: 1411: 1408: 1407: 1406: 1403: 1401: 1398: 1396: 1393: 1391: 1390:Gelfand–Mazur 1388: 1386: 1383: 1381: 1378: 1376: 1373: 1371: 1368: 1366: 1363: 1361: 1358: 1356: 1353: 1351: 1348: 1346: 1343: 1341: 1338: 1336: 1333: 1331: 1328: 1327: 1325: 1323: 1319: 1313: 1310: 1308: 1305: 1301: 1298: 1296: 1293: 1292: 1291: 1288: 1284: 1281: 1279: 1276: 1275: 1274: 1271: 1269: 1266: 1264: 1261: 1259: 1256: 1255: 1253: 1251: 1247: 1241: 1238: 1236: 1233: 1231: 1228: 1226: 1223: 1221: 1218: 1216: 1213: 1211: 1208: 1206: 1203: 1199: 1196: 1195: 1194: 1191: 1189: 1186: 1182: 1179: 1177: 1174: 1173: 1171: 1169: 1166: 1164: 1160: 1156: 1152: 1149: 1148: 1147: 1144: 1142: 1139: 1137: 1133: 1129: 1125: 1122: 1120: 1117: 1115: 1112: 1111: 1110: 1107: 1105: 1102: 1101: 1099: 1097: 1093: 1087: 1084: 1082: 1079: 1075: 1072: 1070: 1067: 1066: 1065: 1062: 1058: 1055: 1053: 1050: 1049: 1048: 1045: 1043: 1040: 1038: 1035: 1031: 1028: 1027: 1026: 1023: 1021: 1018: 1016: 1013: 1012: 1010: 1006: 1000: 997: 995: 992: 988: 985: 984: 983: 980: 978: 975: 973: 970: 966: 962: 959: 958: 957: 954: 950: 947: 946: 945: 942: 940: 937: 935: 932: 930: 927: 926: 924: 920: 913: 909: 905: 902: 900: 896: 894: 891: 889:) convex 888: 885: 883: 880: 878: 874: 872: 869: 867: 864: 862: 858: 854: 850: 847: 845: 842: 841: 840: 837: 835: 834:Grothendieck 832: 830: 827: 823: 820: 819: 818: 815: 813: 810: 809: 807: 803: 798: 791: 786: 784: 779: 777: 772: 771: 768: 762: 759: 758: 748: 744: 740: 734: 730: 726: 722: 718: 714: 710: 704: 700: 696: 692: 688: 684: 680: 676: 670: 666: 662: 661: 656: 655:Rudin, Walter 652: 648: 644: 640: 634: 630: 625: 621: 617: 613: 607: 603: 599: 595: 591: 587: 586: 574: 569: 567: 558: 554: 550: 546: 542: 538: 534: 530: 529: 524: 520: 514: 510: 496: 493: 490: 487: 486: 479: 465: 439: 430: 429:weak-* closed 413: 409: 406: 393: 385: 376: 372: 363: 358: 354: 351: 332: 329: 326: 323: 316:. If for all 303: 277: 256: 248: 232: 215: 201: 181: 173: 172:weak topology 157: 137: 117: 97: 89: 73: 57: 49: 47: 43: 39: 38:weak topology 35: 31: 27: 23: 19: 2823:expanding it 2812: 2766:expanding it 2755: 2740: 2692:Balanced set 2666:Distribution 2604:Applications 2457:Krein–Milman 2442:Closed graph 2219:Applications 2140: 2051: 2012: 1959: 1945: 1941: 1928: 1862: 1814: 1701:Linear cone 1694: 1690: 1679:Convex cone 1572:Paley–Wiener 1432:Mackey–Arens 1422:Krein–Milman 1375:Closed range 1370:Closed graph 1340:Banach–Mazur 1220:Self-adjoint 1124:sesquilinear 857:Polynomially 797:Banach space 728: 694: 659: 628: 593: 583:Bibliography 532: 526: 513: 247:Banach space 218: 88:Banach space 59: 55: 25: 15: 2621:Heat kernel 2611:Hardy space 2518:Trace class 2432:Hahn–Banach 2394:Topological 1940:Continuous 1775:Linear span 1760:Convex hull 1740:Affine hull 1599:holomorphic 1535:holomorphic 1515:Derivatives 1405:Hahn–Banach 1345:Banach–Saks 1263:C*-algebras 1230:Trace class 1193:Functionals 1081:Ultrastrong 994:Quasinormed 573:Conway 1990 523:Šmulian, V. 34:compactness 30:convex hull 18:mathematics 2862:Categories 2554:C*-algebra 2369:Properties 1693:), and (Hw 1594:continuous 1530:functional 1278:C*-algebra 1163:Continuous 1025:Dual space 999:Stereotype 977:Metrizable 904:Projective 505:References 130:(that is, 42:Mark Krein 2528:Unbounded 2523:Transpose 2481:Operators 2410:Separable 2405:Reflexive 2390:Algebraic 2376:Barrelled 2152:Sobolev W 2095:Schwartz 2070:∞ 2031:∞ 2027:ℓ 1993:Ω 1979:λ 1837:Σ 1719:Symmetric 1654:Absorbing 1567:regulated 1547:Integrals 1400:Goldstine 1235:Transpose 1172:Fredholm 1042:Ultraweak 1030:Dual norm 961:Seminorms 929:Barrelled 899:Injective 887:Uniformly 861:Reflexive 747:853623322 727:(2006) . 717:840278135 647:144216834 519:Krein, M. 444:′ 407:≤ 398:′ 381:′ 373:∈ 368:′ 355:∩ 282:′ 52:Statement 2730:Category 2542:Algebras 2424:Theorems 2381:Complete 2350:Schwartz 2296:glossary 2088:weighted 1958:Hilbert 1935:Bs space 1805:Examples 1770:Interior 1746:Relative 1724:Zonotope 1703:(subset) 1681:(subset) 1632:Strongly 1611:Lebesgue 1606:Measures 1476:Analysis 1322:Theorems 1273:Spectrum 1198:positive 1181:operator 1119:operator 1109:Bilinear 1074:operator 1057:operator 1037:Operator 934:Complete 882:Strictly 683:21163277 657:(1991). 620:21195908 592:(1990). 483:See also 403:‖ 390:‖ 2533:Unitary 2513:Nuclear 2498:Compact 2493:Bounded 2488:Adjoint 2462:Min–max 2355:Sobolev 2340:Nuclear 2330:Hilbert 2325:Fréchet 2290: ( 1953:Hardy H 1856:c space 1793:)  1748:)  1669:Bounded 1557:Dunford 1552:Bochner 1525:Gateaux 1520:Fréchet 1295:of ODEs 1240:Unitary 1215:Nuclear 1146:Compact 1136:Bounded 1104:Adjoint 944:Fréchet 939:F-space 910: ( 906:)  859:)  839:Hilbert 812:Asplund 557:0002009 549:1968735 36:in the 2508:Normal 2345:Orlicz 2335:Hölder 2315:Banach 2304:Spaces 2292:topics 1869:Besov 1709:Radial 1674:Convex 1659:Affine 1628:Weakly 1621:Vector 1493:bundle 1283:radius 1210:Normal 1176:kernel 1141:Closed 1064:Strong 982:Normed 972:Mackey 817:Banach 799:topics 745:  735:  715:  705:  681:  671:  645:  635:  618:  608:  555:  547:  24:, the 2813:This 2756:This 2320:Besov 1944:with 1791:Quasi 1785:Polar 1589:Borel 1540:quasi 1069:polar 1052:polar 866:Riesz 545:JSTOR 458:then 245:be a 86:be a 2819:stub 2762:stub 2668:(or 2386:Dual 1942:C(K) 1577:weak 1114:form 1047:Weak 1020:Dual 987:norm 949:tame 822:list 743:OCLC 733:ISBN 713:OCLC 703:ISBN 679:OCLC 669:ISBN 643:OCLC 633:ISBN 616:OCLC 606:ISBN 327:> 249:and 225:Let 90:and 66:Let 44:and 32:and 1159:Dis 699:GTM 537:doi 431:in 427:is 296:of 194:in 16:In 2864:: 2294:– 1929:BV 1863:BK 1815:AC 1697:)) 1630:/ 1132:Un 741:. 711:. 697:. 677:. 667:. 641:. 614:. 604:. 596:. 565:^ 553:MR 551:. 543:. 533:41 521:; 2850:e 2843:t 2836:v 2825:. 2793:e 2786:t 2779:v 2768:. 2672:) 2396:) 2392:/ 2388:( 2298:) 2280:e 2273:t 2266:v 2199:) 2194:p 2190:L 2186:, 2183:X 2180:( 2177:W 2141:F 2122:) 2117:n 2112:R 2107:( 2103:S 2066:L 2052:L 2013:ℓ 1996:) 1990:( 1985:p 1982:, 1975:L 1960:H 1946:K 1906:) 1902:R 1898:( 1893:s 1888:q 1885:, 1882:p 1878:B 1840:) 1834:( 1831:a 1828:b 1789:( 1744:( 1695:x 1691:x 1161:) 1157:( 1134:) 1130:( 963:/ 914:) 897:( 877:B 875:( 855:( 789:e 782:t 775:v 749:. 719:. 685:. 649:. 622:. 559:. 539:: 466:A 440:X 414:} 410:r 394:x 386:: 377:X 364:x 359:{ 352:A 333:, 330:0 324:r 304:X 278:X 257:A 233:X 202:X 182:K 158:X 138:K 118:X 98:K 74:X

Index

mathematics
functional analysis
convex hull
compactness
weak topology
Mark Krein
Vitold Shmulyan
Banach space
weak topology
Banach space
weak-* closed
Krein–Milman theorem
Weak-* topology
Krein, M.
Šmulian, V.
Annals of Mathematics
doi
10.2307/1968735
JSTOR
1968735
MR
0002009


Conway 1990
Conway, John B.
Graduate Texts in Mathematics
Springer-Verlag
ISBN
978-0-387-97245-9

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