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Continuous functions on a compact Hausdorff space

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is not in general a Banach space with respect to the uniform norm since it may contain unbounded functions. Hence it is more typical to consider the space, denoted here
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with respect to the pointwise addition of functions and scalar multiplication by constants. It is, moreover, a
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This is a Banach space (in fact a commutative Banach algebra with identity) with respect to the uniform norm. (
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Hausdorff space. In this case, it is possible to identify a pair of distinguished subsets of
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is an infinite space (since it separates points). Hence, in particular, it is generally not
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In the case of complex functions, the statement holds with the additional hypothesis that
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of real or complex-valued continuous functions can be defined on any topological space
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implies that any normed space is isometrically isomorphic to a subspace of
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Then there is a one-to-one correspondence between Δ and the points of
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can be identified with the collection of all complex homomorphisms
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of a measure, is also a Banach space belonging to the class of
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that contains all constants and separates points, then the
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of algebras which commutes with complex conjugation, then
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by a different form of the Riesz representation theorem. (
1146:{\displaystyle F:{\mathcal {C}}(X)\to {\mathcal {C}}(Y)} 3954: 3880: 3842: 3803: 3751: 3654: 3605: 2424: 2388: 2343: 2298: 2272: 2243: 2214: 2175: 2136: 2097: 2054: 2030: 1996: 1960: 1931: 1908: 1879: 1848: 1819: 1777: 1746: 1713: 1680: 1640: 1613: 1576: 1552: 1508: 1488: 1465: 1429: 1405: 1378: 1358: 1329: 1300: 1265: 1203: 1183: 1163: 1101: 1081: 1061: 1034: 998: 978: 941: 917: 881: 834: 793: 773: 722: 702: 672: 632: 579: 555: 515: 480: 447: 398: 359: 327: 307: 271: 217: 194: 112: 68: 40: 4458: 4382: 4361: 4320: 4259: 4201: 4147: 4082: 3997: 3582: 3511: 3420: 3254: 3099: 3027: 2873: 2786: 2700: 2583: 1627:
is homeomorphic to Δ equipped with this topology. (
4395:Spectral theory of ordinary differential equations 3982: 3906: 3855: 3816: 3779: 3689: 3623: 2449: 2410: 2364: 2329: 2284: 2258: 2229: 2200: 2151: 2122: 2076: 2036: 2005: 1982: 1946: 1917: 1894: 1857: 1834: 1799: 1752: 1732: 1699: 1659: 1619: 1595: 1558: 1539:{\displaystyle {\mathcal {C}}(X)\to \mathbb {C} .} 1538: 1494: 1474: 1451: 1411: 1384: 1364: 1344: 1315: 1286: 1251: 1189: 1169: 1145: 1087: 1067: 1040: 1020: 984: 960: 923: 903: 856: 812: 779: 741: 708: 681: 651: 600: 561: 537: 486: 466: 431: 384: 345: 313: 290: 236: 203: 168: 90: 46: 126: 2085: 2014: 617: 24:, a fundamental role is played by the space of 3907:{\displaystyle S\left(\mathbb {R} ^{n}\right)} 545:Specifically, this dual space is the space of 4051: 4015:Mathematical formulation of quantum mechanics 2560: 2457:In particular, the latter is a Banach space. 8: 2237:consisting of functions such that for every 1674:if and only if it is (uniformly) bounded in 503:Riesz–Markov–Kakutani representation theorem 119: 113: 2020:It is sometimes desirable, particularly in 169:{\displaystyle \|f\|=\sup _{x\in X}|f(x)|,} 4086: 4058: 4044: 4036: 2567: 2553: 2545: 1707:and pointwise convergent. In particular, 3971: 3953: 3894: 3890: 3889: 3879: 3847: 3841: 3808: 3802: 3756: 3750: 3690:{\displaystyle B_{p,q}^{s}(\mathbb {R} )} 3680: 3679: 3670: 3659: 3653: 3604: 2429: 2423: 2393: 2387: 2342: 2316: 2299: 2297: 2271: 2242: 2213: 2180: 2174: 2163:. This is called the space of functions 2135: 2102: 2096: 2059: 2053: 2029: 1995: 1965: 1959: 1930: 1907: 1878: 1847: 1818: 1779: 1778: 1776: 1745: 1715: 1714: 1712: 1682: 1681: 1679: 1642: 1641: 1639: 1612: 1578: 1577: 1575: 1551: 1529: 1528: 1510: 1509: 1507: 1487: 1464: 1431: 1430: 1428: 1404: 1377: 1357: 1328: 1299: 1264: 1202: 1182: 1162: 1128: 1127: 1109: 1108: 1100: 1080: 1060: 1033: 1000: 999: 997: 977: 943: 942: 940: 916: 883: 882: 880: 836: 835: 833: 795: 794: 792: 772: 724: 723: 721: 701: 671: 634: 633: 631: 578: 554: 517: 516: 514: 479: 449: 448: 446: 397: 367: 366: 358: 326: 306: 273: 272: 270: 219: 218: 216: 193: 158: 141: 129: 111: 70: 69: 67: 39: 4348:Group algebra of a locally compact group 601:{\displaystyle \operatorname {rca} (X).} 3780:{\displaystyle L^{\lambda ,p}(\Omega )} 2477:Hewitt, Edwin; Stromberg, Karl (1965), 2165:vanishing in a neighborhood of infinity 608:This space, with the norm given by the 4020:Ordinary Differential Equations (ODEs) 3134:Banach–Steinhaus (Uniform boundedness) 2372:This is called the space of functions 1095:are two compact Hausdorff spaces, and 385:{\displaystyle f\in {\mathcal {C}}(X)} 353:are distinct points, then there is an 1628: 690: 249: 7: 2498:McGraw-Hill Science/Engineering/Math 2467:Dunford, N.; Schwartz, J.T. (1958), 2365:{\displaystyle x\in X\backslash K.} 1990:of bounded continuous functions on 3848: 3809: 3771: 3615: 1925:In the non-compact case, however, 1800:{\displaystyle {\mathcal {C}}(X).} 1570:with respect to this pairing with 1553: 1489: 1452:{\displaystyle {\mathcal {C}}(X).} 1406: 1021:{\displaystyle {\mathcal {C}}(X).} 911:In the case of real functions, if 904:{\displaystyle {\mathcal {C}}(X).} 857:{\displaystyle {\mathcal {C}}(X),} 538:{\displaystyle {\mathcal {C}}(X).} 91:{\displaystyle {\mathcal {C}}(X),} 14: 3512:Subsets / set operations 3289:Differentiation in FrĂ©chet spaces 1733:{\displaystyle {\mathcal {C}}(X)} 1700:{\displaystyle {\mathcal {C}}(X)} 1660:{\displaystyle {\mathcal {C}}(X)} 1596:{\displaystyle {\mathcal {C}}(X)} 1352:are isomorphic as algebras, then 961:{\displaystyle {\mathcal {C}}(X)} 813:{\displaystyle {\mathcal {C}}(X)} 742:{\displaystyle {\mathcal {C}}(X)} 652:{\displaystyle {\mathcal {C}}(X)} 474:is infinite-dimensional whenever 467:{\displaystyle {\mathcal {C}}(X)} 291:{\displaystyle {\mathcal {C}}(X)} 237:{\displaystyle {\mathcal {C}}(X)} 4504: 4503: 4430:Topological quantum field theory 505:gives a characterization of the 180:. The uniform norm defines the 3817:{\displaystyle \ell ^{\infty }} 1252:{\displaystyle F(h)(y)=h(f(y))} 4540:Theory of continuous functions 3977: 3958: 3774: 3768: 3684: 3676: 3618: 3612: 3206:Lomonosov's invariant subspace 3129:Banach–Schauder (open mapping) 2441: 2435: 2405: 2399: 2317: 2313: 2307: 2300: 2224: 2218: 2192: 2186: 2146: 2140: 2114: 2108: 2071: 2065: 1977: 1971: 1941: 1935: 1889: 1883: 1829: 1823: 1791: 1785: 1727: 1721: 1694: 1688: 1654: 1648: 1590: 1584: 1525: 1522: 1516: 1443: 1437: 1339: 1333: 1310: 1304: 1275: 1246: 1243: 1237: 1231: 1222: 1216: 1213: 1207: 1140: 1134: 1124: 1121: 1115: 1012: 1006: 955: 949: 895: 889: 848: 842: 807: 801: 736: 730: 646: 640: 592: 586: 529: 523: 461: 455: 432:{\displaystyle f(x)\neq f(y).} 423: 417: 408: 402: 379: 373: 285: 279: 231: 225: 159: 155: 149: 142: 82: 76: 1: 4226:Uniform boundedness principle 2159:consisting of functions with 1259:for some continuous function 1177:is continuous. Furthermore, 3091:Singular value decomposition 2285:{\displaystyle K\subseteq X} 1740:is only weakly complete for 3856:{\displaystyle L^{\infty }} 3624:{\displaystyle ba(\Sigma )} 3493:Radially convex/Star-shaped 2330:{\displaystyle |f(x)|<r} 2086:Hewitt & Stromberg 1965 2015:Hewitt & Stromberg 1965 624:Positive linear functionals 618:Dunford & Schwartz 1958 248:with respect to this norm.( 4571: 4369:Invariant subspace problem 3983:{\displaystyle W(X,L^{p})} 2479:Real and abstract analysis 2123:{\displaystyle C_{00}(X),} 62:. This space, denoted by 4499: 4089: 3529:Algebraic interior (core) 3144:Cauchy–Schwarz inequality 2787:Function space Topologies 2528:Real and complex analysis 2450:{\displaystyle C_{0}(X).} 2411:{\displaystyle C_{00}(X)} 2201:{\displaystyle C_{0}(X),} 1287:{\displaystyle f:Y\to X.} 873:Stone–Weierstrass theorem 659:correspond to (positive) 4338:Spectrum of a C*-algebra 2469:Linear operators, Part I 2077:{\displaystyle C_{B}(X)} 1983:{\displaystyle C_{B}(X)} 346:{\displaystyle x,y\in X} 4435:Noncommutative geometry 2266:there is a compact set 2259:{\displaystyle r>0,} 1559:{\displaystyle \Delta } 1495:{\displaystyle \Delta } 1412:{\displaystyle \Delta } 4491:Tomita–Takesaki theory 4466:Approximation property 4410:Calculus of variations 3984: 3908: 3857: 3818: 3781: 3691: 3625: 2794:Banach–Mazur compactum 2584:Types of Banach spaces 2451: 2412: 2366: 2331: 2286: 2260: 2231: 2202: 2153: 2124: 2078: 2038: 2007: 1984: 1948: 1919: 1896: 1859: 1836: 1811:Banach–Alaoglu theorem 1801: 1754: 1734: 1701: 1661: 1621: 1597: 1560: 1540: 1496: 1476: 1453: 1413: 1386: 1366: 1346: 1317: 1288: 1253: 1191: 1171: 1147: 1089: 1069: 1042: 1022: 986: 962: 925: 905: 858: 814: 781: 743: 710: 683: 653: 602: 563: 539: 488: 468: 433: 386: 347: 315: 292: 238: 205: 170: 92: 48: 4486:Banach–Mazur distance 4449:Generalized functions 4010:Finite element method 4005:Differential operator 3985: 3909: 3858: 3819: 3782: 3692: 3626: 3466:Convex series related 3262:Abstract Wiener space 3189:hyperplane separation 2744:Minkowski functionals 2628:Polarization identity 2452: 2413: 2375:vanishing at infinity 2367: 2332: 2287: 2261: 2232: 2203: 2154: 2125: 2079: 2039: 2008: 1985: 1949: 1920: 1897: 1860: 1837: 1802: 1755: 1735: 1702: 1662: 1622: 1598: 1561: 1541: 1497: 1477: 1454: 1414: 1387: 1367: 1347: 1318: 1289: 1254: 1192: 1172: 1148: 1090: 1070: 1043: 1023: 987: 963: 926: 906: 859: 824:if and only if it is 815: 782: 765:Arzelà–Ascoli theorem 744: 711: 684: 654: 603: 564: 540: 507:continuous dual space 489: 469: 434: 387: 348: 316: 293: 239: 206: 171: 106:with norm defined by 93: 49: 18:mathematical analysis 4231:Kakutani fixed-point 4216:Riesz representation 3952: 3878: 3840: 3801: 3749: 3652: 3603: 3592:Absolute continuity 3246:Schauder fixed-point 3236:Riesz representation 3196:Kakutani fixed-point 3164:Freudenthal spectral 2650:L-semi-inner product 2471:, Wiley-Interscience 2422: 2386: 2341: 2296: 2270: 2241: 2230:{\displaystyle C(X)} 2212: 2173: 2152:{\displaystyle C(X)} 2134: 2095: 2052: 2028: 1994: 1958: 1947:{\displaystyle C(X)} 1929: 1906: 1895:{\displaystyle C(X)} 1877: 1846: 1835:{\displaystyle C(X)} 1817: 1775: 1744: 1711: 1678: 1638: 1611: 1574: 1550: 1506: 1486: 1463: 1427: 1403: 1376: 1356: 1345:{\displaystyle C(Y)} 1327: 1316:{\displaystyle C(X)} 1298: 1263: 1201: 1181: 1161: 1099: 1079: 1059: 1032: 996: 976: 939: 915: 879: 832: 791: 771: 720: 700: 670: 630: 577: 553: 513: 478: 445: 396: 357: 325: 305: 269: 215: 192: 110: 66: 38: 26:continuous functions 4545:Functional analysis 4415:Functional calculus 4374:Mahler's conjecture 4353:Von Neumann algebra 4067:Functional analysis 3675: 3413:measurable function 3363:Functional calculus 3226:Parseval's identity 3139:Bessel's inequality 3086:Polar decomposition 2865:Uniform convergence 2623:Inner product space 2493:Functional Analysis 1396:topological spaces. 1050:complex conjugation 186:uniform convergence 54:with values in the 22:functional analysis 4555:Types of functions 4440:Riemann hypothesis 4139:Topological vector 4025:Validated numerics 3980: 3936:Sobolev inequality 3904: 3853: 3814: 3777: 3706:Bounded variation 3687: 3655: 3640:Banach coordinate 3621: 3559:Minkowski addition 3221:M. Riesz extension 2701:Banach spaces are: 2447: 2408: 2362: 2327: 2282: 2256: 2227: 2198: 2149: 2120: 2074: 2034: 2006:{\displaystyle X.} 2003: 1980: 1944: 1918:{\displaystyle X.} 1915: 1892: 1858:{\displaystyle X.} 1855: 1832: 1797: 1750: 1730: 1697: 1657: 1617: 1593: 1556: 1536: 1492: 1475:{\displaystyle X.} 1472: 1449: 1409: 1382: 1362: 1342: 1313: 1294:In particular, if 1284: 1249: 1187: 1167: 1143: 1085: 1065: 1038: 1018: 982: 958: 921: 901: 854: 822:relatively compact 810: 777: 739: 716:is infinite, then 706: 682:{\displaystyle X,} 679: 649: 598: 559: 535: 484: 464: 429: 382: 343: 311: 288: 234: 204:{\displaystyle X.} 201: 166: 140: 88: 44: 4517: 4516: 4420:Integral operator 4197: 4196: 4033: 4032: 3745:Morrey–Campanato 3727:compact Hausdorff 3574:Relative interior 3428:Absolutely convex 3395:Projection-valued 3004:Strictly singular 2930:on Hilbert spaces 2691:of Hilbert spaces 2507:978-0-07-054236-5 2481:, Springer-Verlag 2037:{\displaystyle X} 1753:{\displaystyle X} 1620:{\displaystyle X} 1605:Gelfand transform 1385:{\displaystyle Y} 1365:{\displaystyle X} 1190:{\displaystyle F} 1170:{\displaystyle F} 1088:{\displaystyle Y} 1068:{\displaystyle X} 1041:{\displaystyle A} 985:{\displaystyle A} 924:{\displaystyle A} 780:{\displaystyle K} 709:{\displaystyle X} 562:{\displaystyle X} 487:{\displaystyle X} 314:{\displaystyle X} 125: 47:{\displaystyle X} 20:, and especially 4562: 4535:Complex analysis 4507: 4506: 4425:Jones polynomial 4343:Operator algebra 4087: 4060: 4053: 4046: 4037: 3989: 3987: 3986: 3981: 3976: 3975: 3943:Triebel–Lizorkin 3913: 3911: 3910: 3905: 3903: 3899: 3898: 3893: 3862: 3860: 3859: 3854: 3852: 3851: 3823: 3821: 3820: 3815: 3813: 3812: 3786: 3784: 3783: 3778: 3767: 3766: 3696: 3694: 3693: 3688: 3683: 3674: 3669: 3630: 3628: 3627: 3622: 3483: 3461: 3443:Balanced/Circled 3241:Robinson-Ursescu 3159:Eberlein–Šmulian 3079:Spectral theorem 2875:Linear operators 2672:Uniformly smooth 2569: 2562: 2555: 2546: 2540: 2519: 2482: 2472: 2456: 2454: 2453: 2448: 2434: 2433: 2417: 2415: 2414: 2409: 2398: 2397: 2371: 2369: 2368: 2363: 2336: 2334: 2333: 2328: 2320: 2303: 2291: 2289: 2288: 2283: 2265: 2263: 2262: 2257: 2236: 2234: 2233: 2228: 2207: 2205: 2204: 2199: 2185: 2184: 2158: 2156: 2155: 2150: 2129: 2127: 2126: 2121: 2107: 2106: 2083: 2081: 2080: 2075: 2064: 2063: 2043: 2041: 2040: 2035: 2012: 2010: 2009: 2004: 1989: 1987: 1986: 1981: 1970: 1969: 1953: 1951: 1950: 1945: 1924: 1922: 1921: 1916: 1901: 1899: 1898: 1893: 1864: 1862: 1861: 1856: 1841: 1839: 1838: 1833: 1806: 1804: 1803: 1798: 1784: 1783: 1759: 1757: 1756: 1751: 1739: 1737: 1736: 1731: 1720: 1719: 1706: 1704: 1703: 1698: 1687: 1686: 1666: 1664: 1663: 1658: 1647: 1646: 1626: 1624: 1623: 1618: 1602: 1600: 1599: 1594: 1583: 1582: 1568:initial topology 1565: 1563: 1562: 1557: 1545: 1543: 1542: 1537: 1532: 1515: 1514: 1501: 1499: 1498: 1493: 1481: 1479: 1478: 1473: 1458: 1456: 1455: 1450: 1436: 1435: 1419:be the space of 1418: 1416: 1415: 1410: 1391: 1389: 1388: 1383: 1371: 1369: 1368: 1363: 1351: 1349: 1348: 1343: 1322: 1320: 1319: 1314: 1293: 1291: 1290: 1285: 1258: 1256: 1255: 1250: 1196: 1194: 1193: 1188: 1176: 1174: 1173: 1168: 1152: 1150: 1149: 1144: 1133: 1132: 1114: 1113: 1094: 1092: 1091: 1086: 1074: 1072: 1071: 1066: 1048:is closed under 1047: 1045: 1044: 1039: 1027: 1025: 1024: 1019: 1005: 1004: 991: 989: 988: 983: 967: 965: 964: 959: 948: 947: 930: 928: 927: 922: 910: 908: 907: 902: 888: 887: 863: 861: 860: 855: 841: 840: 819: 817: 816: 811: 800: 799: 786: 784: 783: 778: 767:holds: A subset 748: 746: 745: 740: 729: 728: 715: 713: 712: 707: 688: 686: 685: 680: 658: 656: 655: 650: 639: 638: 607: 605: 604: 599: 568: 566: 565: 560: 544: 542: 541: 536: 522: 521: 493: 491: 490: 485: 473: 471: 470: 465: 454: 453: 438: 436: 435: 430: 391: 389: 388: 383: 372: 371: 352: 350: 349: 344: 320: 318: 317: 312: 299:separates points 297: 295: 294: 289: 278: 277: 243: 241: 240: 235: 224: 223: 210: 208: 207: 202: 188:of functions on 175: 173: 172: 167: 162: 145: 139: 97: 95: 94: 89: 75: 74: 53: 51: 50: 45: 4570: 4569: 4565: 4564: 4563: 4561: 4560: 4559: 4520: 4519: 4518: 4513: 4495: 4459:Advanced topics 4454: 4378: 4357: 4316: 4282:Hilbert–Schmidt 4255: 4246:Gelfand–Naimark 4193: 4143: 4078: 4064: 4034: 4029: 3993: 3967: 3950: 3949: 3948:Wiener amalgam 3918:Segal–Bargmann 3888: 3884: 3876: 3875: 3843: 3838: 3837: 3804: 3799: 3798: 3752: 3747: 3746: 3701:Birnbaum–Orlicz 3650: 3649: 3601: 3600: 3578: 3534:Bounding points 3507: 3481: 3459: 3416: 3267:Banach manifold 3250: 3174:Gelfand–Naimark 3095: 3069:Spectral theory 3037:Banach algebras 3029:Operator theory 3023: 2984:Pseudo-monotone 2967:Hilbert–Schmidt 2947:Densely defined 2869: 2782: 2696: 2579: 2573: 2538: 2530:, McGraw-Hill, 2522: 2508: 2486: 2476: 2466: 2463: 2425: 2420: 2419: 2389: 2384: 2383: 2382:The closure of 2339: 2338: 2294: 2293: 2268: 2267: 2239: 2238: 2210: 2209: 2176: 2171: 2170: 2161:compact support 2132: 2131: 2098: 2093: 2092: 2055: 2050: 2049: 2046:locally compact 2026: 2025: 2017:, Theorem 7.9) 1992: 1991: 1961: 1956: 1955: 1927: 1926: 1904: 1903: 1875: 1874: 1871: 1869:Generalizations 1844: 1843: 1815: 1814: 1773: 1772: 1771:on the dual of 1742: 1741: 1709: 1708: 1676: 1675: 1636: 1635: 1609: 1608: 1572: 1571: 1548: 1547: 1504: 1503: 1484: 1483: 1461: 1460: 1425: 1424: 1401: 1400: 1374: 1373: 1354: 1353: 1325: 1324: 1296: 1295: 1261: 1260: 1199: 1198: 1179: 1178: 1159: 1158: 1097: 1096: 1077: 1076: 1057: 1056: 1030: 1029: 994: 993: 974: 973: 937: 936: 913: 912: 877: 876: 830: 829: 828:in the norm of 789: 788: 769: 768: 718: 717: 698: 697: 668: 667: 628: 627: 610:total variation 575: 574: 551: 550: 511: 510: 496:locally compact 476: 475: 443: 442: 394: 393: 355: 354: 323: 322: 303: 302: 267: 266: 263:Urysohn's lemma 258: 213: 212: 190: 189: 108: 107: 64: 63: 60:complex numbers 36: 35: 33:Hausdorff space 12: 11: 5: 4568: 4566: 4558: 4557: 4552: 4547: 4542: 4537: 4532: 4522: 4521: 4515: 4514: 4512: 4511: 4500: 4497: 4496: 4494: 4493: 4488: 4483: 4478: 4476:Choquet theory 4473: 4468: 4462: 4460: 4456: 4455: 4453: 4452: 4442: 4437: 4432: 4427: 4422: 4417: 4412: 4407: 4402: 4397: 4392: 4386: 4384: 4380: 4379: 4377: 4376: 4371: 4365: 4363: 4359: 4358: 4356: 4355: 4350: 4345: 4340: 4335: 4330: 4328:Banach algebra 4324: 4322: 4318: 4317: 4315: 4314: 4309: 4304: 4299: 4294: 4289: 4284: 4279: 4274: 4269: 4263: 4261: 4257: 4256: 4254: 4253: 4251:Banach–Alaoglu 4248: 4243: 4238: 4233: 4228: 4223: 4218: 4213: 4207: 4205: 4199: 4198: 4195: 4194: 4192: 4191: 4186: 4181: 4179:Locally convex 4176: 4162: 4157: 4151: 4149: 4145: 4144: 4142: 4141: 4136: 4131: 4126: 4121: 4116: 4111: 4106: 4101: 4096: 4090: 4084: 4080: 4079: 4065: 4063: 4062: 4055: 4048: 4040: 4031: 4030: 4028: 4027: 4022: 4017: 4012: 4007: 4001: 3999: 3995: 3994: 3992: 3991: 3979: 3974: 3970: 3966: 3963: 3960: 3957: 3945: 3940: 3939: 3938: 3928: 3926:Sequence space 3923: 3915: 3902: 3897: 3892: 3887: 3883: 3871: 3870: 3869: 3864: 3850: 3846: 3827: 3826: 3825: 3811: 3807: 3788: 3776: 3773: 3770: 3765: 3762: 3759: 3755: 3742: 3734: 3729: 3716: 3711: 3703: 3698: 3686: 3682: 3678: 3673: 3668: 3665: 3662: 3658: 3645: 3637: 3632: 3620: 3617: 3614: 3611: 3608: 3597: 3588: 3586: 3580: 3579: 3577: 3576: 3566: 3561: 3556: 3551: 3546: 3541: 3536: 3531: 3521: 3515: 3513: 3509: 3508: 3506: 3505: 3500: 3495: 3490: 3485: 3477: 3463: 3455: 3450: 3445: 3440: 3435: 3430: 3424: 3422: 3418: 3417: 3415: 3414: 3404: 3403: 3402: 3397: 3392: 3382: 3381: 3380: 3375: 3370: 3360: 3359: 3358: 3353: 3348: 3343: 3341:Gelfand–Pettis 3338: 3333: 3323: 3322: 3321: 3316: 3311: 3306: 3301: 3291: 3286: 3281: 3276: 3275: 3274: 3264: 3258: 3256: 3252: 3251: 3249: 3248: 3243: 3238: 3233: 3228: 3223: 3218: 3213: 3208: 3203: 3198: 3193: 3192: 3191: 3181: 3176: 3171: 3166: 3161: 3156: 3151: 3146: 3141: 3136: 3131: 3126: 3121: 3116: 3114:Banach–Alaoglu 3111: 3109:Anderson–Kadec 3105: 3103: 3097: 3096: 3094: 3093: 3088: 3083: 3082: 3081: 3076: 3066: 3065: 3064: 3059: 3049: 3047:Operator space 3044: 3039: 3033: 3031: 3025: 3024: 3022: 3021: 3016: 3011: 3006: 3001: 2996: 2991: 2986: 2981: 2980: 2979: 2969: 2964: 2963: 2962: 2957: 2949: 2944: 2934: 2933: 2932: 2922: 2917: 2907: 2906: 2905: 2900: 2895: 2885: 2879: 2877: 2871: 2870: 2868: 2867: 2862: 2857: 2856: 2855: 2850: 2840: 2839: 2838: 2833: 2823: 2818: 2813: 2812: 2811: 2801: 2796: 2790: 2788: 2784: 2783: 2781: 2780: 2775: 2770: 2769: 2768: 2758: 2753: 2748: 2747: 2746: 2735:Locally convex 2732: 2731: 2730: 2720: 2715: 2710: 2704: 2702: 2698: 2697: 2695: 2694: 2687:Tensor product 2680: 2674: 2669: 2663: 2658: 2652: 2647: 2642: 2632: 2631: 2630: 2625: 2615: 2610: 2608:Banach lattice 2605: 2604: 2603: 2593: 2587: 2585: 2581: 2580: 2574: 2572: 2571: 2564: 2557: 2549: 2543: 2542: 2536: 2520: 2506: 2484: 2474: 2462: 2459: 2446: 2443: 2440: 2437: 2432: 2428: 2407: 2404: 2401: 2396: 2392: 2380: 2379: 2361: 2358: 2355: 2352: 2349: 2346: 2326: 2323: 2319: 2315: 2312: 2309: 2306: 2302: 2281: 2278: 2275: 2255: 2252: 2249: 2246: 2226: 2223: 2220: 2217: 2208:the subset of 2197: 2194: 2191: 2188: 2183: 2179: 2168: 2148: 2145: 2142: 2139: 2130:the subset of 2119: 2116: 2113: 2110: 2105: 2101: 2073: 2070: 2067: 2062: 2058: 2033: 2022:measure theory 2002: 1999: 1979: 1976: 1973: 1968: 1964: 1943: 1940: 1937: 1934: 1914: 1911: 1891: 1888: 1885: 1882: 1870: 1867: 1866: 1865: 1854: 1851: 1831: 1828: 1825: 1822: 1807: 1796: 1793: 1790: 1787: 1782: 1769:weak* topology 1765:vague topology 1761: 1749: 1729: 1726: 1723: 1718: 1696: 1693: 1690: 1685: 1656: 1653: 1650: 1645: 1634:A sequence in 1632: 1616: 1603:(that is, the 1592: 1589: 1586: 1581: 1555: 1535: 1531: 1527: 1524: 1521: 1518: 1513: 1491: 1471: 1468: 1448: 1445: 1442: 1439: 1434: 1421:maximal ideals 1408: 1397: 1381: 1361: 1341: 1338: 1335: 1332: 1312: 1309: 1306: 1303: 1283: 1280: 1277: 1274: 1271: 1268: 1248: 1245: 1242: 1239: 1236: 1233: 1230: 1227: 1224: 1221: 1218: 1215: 1212: 1209: 1206: 1186: 1166: 1142: 1139: 1136: 1131: 1126: 1123: 1120: 1117: 1112: 1107: 1104: 1084: 1064: 1053: 1037: 1017: 1014: 1011: 1008: 1003: 981: 957: 954: 951: 946: 920: 900: 897: 894: 891: 886: 869: 866:equicontinuous 853: 850: 847: 844: 839: 809: 806: 803: 798: 776: 761: 738: 735: 732: 727: 705: 694: 678: 675: 664:Borel measures 648: 645: 642: 637: 621: 597: 594: 591: 588: 585: 582: 573:), denoted by 571:Borel measures 558: 547:Radon measures 534: 531: 528: 525: 520: 499: 483: 463: 460: 457: 452: 439: 428: 425: 422: 419: 416: 413: 410: 407: 404: 401: 381: 378: 375: 370: 365: 362: 342: 339: 336: 333: 330: 310: 287: 284: 281: 276: 257: 254: 246:Banach algebra 233: 230: 227: 222: 200: 197: 165: 161: 157: 154: 151: 148: 144: 138: 135: 132: 128: 124: 121: 118: 115: 87: 84: 81: 78: 73: 43: 13: 10: 9: 6: 4: 3: 2: 4567: 4556: 4553: 4551: 4550:Real analysis 4548: 4546: 4543: 4541: 4538: 4536: 4533: 4531: 4530:Banach spaces 4528: 4527: 4525: 4510: 4502: 4501: 4498: 4492: 4489: 4487: 4484: 4482: 4481:Weak topology 4479: 4477: 4474: 4472: 4469: 4467: 4464: 4463: 4461: 4457: 4450: 4446: 4443: 4441: 4438: 4436: 4433: 4431: 4428: 4426: 4423: 4421: 4418: 4416: 4413: 4411: 4408: 4406: 4405:Index theorem 4403: 4401: 4398: 4396: 4393: 4391: 4388: 4387: 4385: 4381: 4375: 4372: 4370: 4367: 4366: 4364: 4362:Open problems 4360: 4354: 4351: 4349: 4346: 4344: 4341: 4339: 4336: 4334: 4331: 4329: 4326: 4325: 4323: 4319: 4313: 4310: 4308: 4305: 4303: 4300: 4298: 4295: 4293: 4290: 4288: 4285: 4283: 4280: 4278: 4275: 4273: 4270: 4268: 4265: 4264: 4262: 4258: 4252: 4249: 4247: 4244: 4242: 4239: 4237: 4234: 4232: 4229: 4227: 4224: 4222: 4219: 4217: 4214: 4212: 4209: 4208: 4206: 4204: 4200: 4190: 4187: 4185: 4182: 4180: 4177: 4174: 4170: 4166: 4163: 4161: 4158: 4156: 4153: 4152: 4150: 4146: 4140: 4137: 4135: 4132: 4130: 4127: 4125: 4122: 4120: 4117: 4115: 4112: 4110: 4107: 4105: 4102: 4100: 4097: 4095: 4092: 4091: 4088: 4085: 4081: 4076: 4072: 4068: 4061: 4056: 4054: 4049: 4047: 4042: 4041: 4038: 4026: 4023: 4021: 4018: 4016: 4013: 4011: 4008: 4006: 4003: 4002: 4000: 3996: 3990: 3972: 3968: 3964: 3961: 3955: 3946: 3944: 3941: 3937: 3934: 3933: 3932: 3929: 3927: 3924: 3922: 3921: 3916: 3914: 3900: 3895: 3885: 3881: 3872: 3868: 3865: 3863: 3844: 3835: 3834: 3833: 3832: 3828: 3824: 3805: 3796: 3795: 3794: 3793: 3789: 3787: 3763: 3760: 3757: 3753: 3743: 3741: 3740: 3735: 3733: 3730: 3728: 3726: 3722: 3717: 3715: 3712: 3710: 3709: 3704: 3702: 3699: 3697: 3671: 3666: 3663: 3660: 3656: 3646: 3644: 3643: 3638: 3636: 3633: 3631: 3609: 3606: 3598: 3596: 3595: 3590: 3589: 3587: 3585: 3581: 3575: 3571: 3567: 3565: 3562: 3560: 3557: 3555: 3552: 3550: 3547: 3545: 3544:Extreme point 3542: 3540: 3537: 3535: 3532: 3530: 3526: 3522: 3520: 3517: 3516: 3514: 3510: 3504: 3501: 3499: 3496: 3494: 3491: 3489: 3486: 3484: 3478: 3475: 3471: 3467: 3464: 3462: 3456: 3454: 3451: 3449: 3446: 3444: 3441: 3439: 3436: 3434: 3431: 3429: 3426: 3425: 3423: 3421:Types of sets 3419: 3412: 3408: 3405: 3401: 3398: 3396: 3393: 3391: 3388: 3387: 3386: 3383: 3379: 3376: 3374: 3371: 3369: 3366: 3365: 3364: 3361: 3357: 3354: 3352: 3349: 3347: 3344: 3342: 3339: 3337: 3334: 3332: 3329: 3328: 3327: 3324: 3320: 3317: 3315: 3312: 3310: 3307: 3305: 3302: 3300: 3297: 3296: 3295: 3292: 3290: 3287: 3285: 3284:Convex series 3282: 3280: 3279:Bochner space 3277: 3273: 3270: 3269: 3268: 3265: 3263: 3260: 3259: 3257: 3253: 3247: 3244: 3242: 3239: 3237: 3234: 3232: 3231:Riesz's lemma 3229: 3227: 3224: 3222: 3219: 3217: 3216:Mazur's lemma 3214: 3212: 3209: 3207: 3204: 3202: 3199: 3197: 3194: 3190: 3187: 3186: 3185: 3182: 3180: 3177: 3175: 3172: 3170: 3169:Gelfand–Mazur 3167: 3165: 3162: 3160: 3157: 3155: 3152: 3150: 3147: 3145: 3142: 3140: 3137: 3135: 3132: 3130: 3127: 3125: 3122: 3120: 3117: 3115: 3112: 3110: 3107: 3106: 3104: 3102: 3098: 3092: 3089: 3087: 3084: 3080: 3077: 3075: 3072: 3071: 3070: 3067: 3063: 3060: 3058: 3055: 3054: 3053: 3050: 3048: 3045: 3043: 3040: 3038: 3035: 3034: 3032: 3030: 3026: 3020: 3017: 3015: 3012: 3010: 3007: 3005: 3002: 3000: 2997: 2995: 2992: 2990: 2987: 2985: 2982: 2978: 2975: 2974: 2973: 2970: 2968: 2965: 2961: 2958: 2956: 2953: 2952: 2950: 2948: 2945: 2943: 2939: 2935: 2931: 2928: 2927: 2926: 2923: 2921: 2918: 2916: 2912: 2908: 2904: 2901: 2899: 2896: 2894: 2891: 2890: 2889: 2886: 2884: 2881: 2880: 2878: 2876: 2872: 2866: 2863: 2861: 2858: 2854: 2851: 2849: 2846: 2845: 2844: 2841: 2837: 2834: 2832: 2829: 2828: 2827: 2824: 2822: 2819: 2817: 2814: 2810: 2807: 2806: 2805: 2802: 2800: 2797: 2795: 2792: 2791: 2789: 2785: 2779: 2776: 2774: 2771: 2767: 2764: 2763: 2762: 2759: 2757: 2754: 2752: 2749: 2745: 2741: 2738: 2737: 2736: 2733: 2729: 2726: 2725: 2724: 2721: 2719: 2716: 2714: 2711: 2709: 2706: 2705: 2703: 2699: 2692: 2688: 2684: 2681: 2679: 2675: 2673: 2670: 2668:) convex 2667: 2664: 2662: 2659: 2657: 2653: 2651: 2648: 2646: 2643: 2641: 2637: 2633: 2629: 2626: 2624: 2621: 2620: 2619: 2616: 2614: 2613:Grothendieck 2611: 2609: 2606: 2602: 2599: 2598: 2597: 2594: 2592: 2589: 2588: 2586: 2582: 2577: 2570: 2565: 2563: 2558: 2556: 2551: 2550: 2547: 2539: 2537:0-07-054234-1 2533: 2529: 2525: 2524:Rudin, Walter 2521: 2517: 2513: 2509: 2503: 2499: 2495: 2494: 2489: 2488:Rudin, Walter 2485: 2480: 2475: 2470: 2465: 2464: 2460: 2458: 2444: 2438: 2430: 2426: 2418:is precisely 2402: 2394: 2390: 2377: 2376: 2359: 2356: 2350: 2347: 2344: 2324: 2321: 2310: 2304: 2279: 2276: 2273: 2253: 2250: 2247: 2244: 2221: 2215: 2195: 2189: 2181: 2177: 2169: 2166: 2162: 2143: 2137: 2117: 2111: 2103: 2099: 2091: 2090: 2089: 2087: 2068: 2060: 2056: 2047: 2031: 2023: 2018: 2016: 2000: 1997: 1974: 1966: 1962: 1938: 1932: 1912: 1909: 1886: 1880: 1868: 1852: 1849: 1826: 1820: 1812: 1808: 1794: 1788: 1770: 1766: 1762: 1760:a finite set. 1747: 1724: 1691: 1673: 1670: 1651: 1633: 1630: 1614: 1606: 1587: 1569: 1533: 1519: 1482:Furthermore, 1469: 1466: 1446: 1440: 1422: 1398: 1395: 1379: 1359: 1336: 1330: 1307: 1301: 1281: 1278: 1272: 1269: 1266: 1240: 1234: 1228: 1225: 1219: 1210: 1204: 1197:has the form 1184: 1164: 1156: 1137: 1118: 1105: 1102: 1082: 1062: 1054: 1051: 1035: 1015: 1009: 979: 971: 952: 934: 918: 898: 892: 874: 870: 867: 851: 845: 827: 823: 804: 774: 766: 762: 759: 756: 752: 733: 703: 695: 692: 676: 673: 665: 662: 643: 625: 622: 619: 615: 611: 595: 589: 583: 580: 572: 556: 548: 532: 526: 508: 504: 500: 497: 481: 458: 440: 426: 420: 414: 411: 405: 399: 376: 363: 360: 340: 337: 334: 331: 328: 308: 300: 282: 264: 260: 259: 255: 253: 251: 247: 228: 198: 195: 187: 183: 179: 163: 152: 146: 136: 133: 130: 122: 116: 105: 101: 85: 79: 61: 57: 41: 34: 31: 27: 23: 19: 4471:Balanced set 4445:Distribution 4383:Applications 4236:Krein–Milman 4221:Closed graph 3998:Applications 3919: 3830: 3791: 3738: 3724: 3720: 3718: 3707: 3641: 3593: 3480:Linear cone 3473: 3469: 3458:Convex cone 3351:Paley–Wiener 3211:Mackey–Arens 3201:Krein–Milman 3154:Closed range 3149:Closed graph 3119:Banach–Mazur 2999:Self-adjoint 2903:sesquilinear 2636:Polynomially 2576:Banach space 2527: 2492: 2478: 2468: 2381: 2373: 2164: 2019: 1872: 1394:homeomorphic 1155:homomorphism 753:, nor is it 693:, Chapter 2) 178:uniform norm 104:normed space 100:vector space 15: 4400:Heat kernel 4390:Hardy space 4297:Trace class 4211:Hahn–Banach 4173:Topological 3719:Continuous 3554:Linear span 3539:Convex hull 3519:Affine hull 3378:holomorphic 3314:holomorphic 3294:Derivatives 3184:Hahn–Banach 3124:Banach–Saks 3042:C*-algebras 3009:Trace class 2972:Functionals 2860:Ultrastrong 2773:Quasinormed 4524:Categories 4333:C*-algebra 4148:Properties 3472:), and (Hw 3373:continuous 3309:functional 3057:C*-algebra 2942:Continuous 2804:Dual space 2778:Stereotype 2756:Metrizable 2683:Projective 2461:References 2292:such that 1873:The space 1629:Rudin 1973 875:holds for 691:Rudin 1966 620:, §IV.6.3) 441:The space 392:such that 256:Properties 250:Rudin 1973 211:The space 4307:Unbounded 4302:Transpose 4260:Operators 4189:Separable 4184:Reflexive 4169:Algebraic 4155:Barrelled 3931:Sobolev W 3874:Schwartz 3849:∞ 3810:∞ 3806:ℓ 3772:Ω 3758:λ 3616:Σ 3498:Symmetric 3433:Absorbing 3346:regulated 3326:Integrals 3179:Goldstine 3014:Transpose 2951:Fredholm 2821:Ultraweak 2809:Dual norm 2740:Seminorms 2708:Barrelled 2678:Injective 2666:Uniformly 2640:Reflexive 2354:∖ 2348:∈ 2277:⊆ 2088:, §II.7) 1842:for some 1631:, §11.13) 1607:). Then 1566:with the 1554:Δ 1526:→ 1490:Δ 1407:Δ 1276:→ 1125:→ 751:reflexive 614:ba spaces 584:⁡ 569:(regular 412:≠ 364:∈ 338:∈ 252:, §11.3) 134:∈ 120:‖ 114:‖ 4509:Category 4321:Algebras 4203:Theorems 4160:Complete 4129:Schwartz 4075:glossary 3867:weighted 3737:Hilbert 3714:Bs space 3584:Examples 3549:Interior 3525:Relative 3503:Zonotope 3482:(subset) 3460:(subset) 3411:Strongly 3390:Lebesgue 3385:Measures 3255:Analysis 3101:Theorems 3052:Spectrum 2977:positive 2960:operator 2898:operator 2888:Bilinear 2853:operator 2836:operator 2816:Operator 2713:Complete 2661:Strictly 2526:(1966), 2516:21163277 2490:(1991). 2337:for all 758:complete 182:topology 4312:Unitary 4292:Nuclear 4277:Compact 4272:Bounded 4267:Adjoint 4241:Min–max 4134:Sobolev 4119:Nuclear 4109:Hilbert 4104:FrĂ©chet 4069: ( 3732:Hardy H 3635:c space 3572:)  3527:)  3448:Bounded 3336:Dunford 3331:Bochner 3304:Gateaux 3299:FrĂ©chet 3074:of ODEs 3019:Unitary 2994:Nuclear 2925:Compact 2915:Bounded 2883:Adjoint 2723:FrĂ©chet 2718:F-space 2689: ( 2685:)  2638:)  2618:Hilbert 2591:Asplund 1767:is the 970:closure 933:subring 826:bounded 749:is not 661:regular 30:compact 4287:Normal 4124:Orlicz 4114:Hölder 4094:Banach 4083:Spaces 4071:topics 3648:Besov 3488:Radial 3453:Convex 3438:Affine 3407:Weakly 3400:Vector 3272:bundle 3062:radius 2989:Normal 2955:kernel 2920:Closed 2843:Strong 2761:Normed 2751:Mackey 2596:Banach 2578:topics 2534:  2514:  2504:  1672:Cauchy 1669:weakly 1546:Equip 755:weakly 4099:Besov 3723:with 3570:Quasi 3564:Polar 3368:Borel 3319:quasi 2848:polar 2831:polar 2645:Riesz 2044:is a 1153:is a 931:is a 321:: If 244:is a 98:is a 28:on a 4447:(or 4165:Dual 3721:C(K) 3356:weak 2893:form 2826:Weak 2799:Dual 2766:norm 2728:tame 2601:list 2532:ISBN 2512:OCLC 2502:ISBN 2322:< 2248:> 1809:The 1763:The 1399:Let 1392:are 1372:and 1323:and 1075:and 871:The 864:and 763:The 501:The 176:the 56:real 2938:Dis 2084:: ( 1667:is 1423:in 1055:If 992:is 972:of 935:of 820:is 787:of 696:If 666:on 626:on 616:. ( 581:rca 549:on 509:of 301:of 261:By 184:of 127:sup 58:or 16:In 4526:: 4073:– 3708:BV 3642:BK 3594:AC 3476:)) 3409:/ 2911:Un 2510:. 2500:. 2395:00 2104:00 265:, 4451:) 4175:) 4171:/ 4167:( 4077:) 4059:e 4052:t 4045:v 3978:) 3973:p 3969:L 3965:, 3962:X 3959:( 3956:W 3920:F 3901:) 3896:n 3891:R 3886:( 3882:S 3845:L 3831:L 3792:â„“ 3775:) 3769:( 3764:p 3761:, 3754:L 3739:H 3725:K 3685:) 3681:R 3677:( 3672:s 3667:q 3664:, 3661:p 3657:B 3619:) 3613:( 3610:a 3607:b 3568:( 3523:( 3474:x 3470:x 2940:) 2936:( 2913:) 2909:( 2742:/ 2693:) 2676:( 2656:B 2654:( 2634:( 2568:e 2561:t 2554:v 2541:. 2518:. 2483:. 2473:. 2445:. 2442:) 2439:X 2436:( 2431:0 2427:C 2406:) 2403:X 2400:( 2391:C 2378:. 2360:. 2357:K 2351:X 2345:x 2325:r 2318:| 2314:) 2311:x 2308:( 2305:f 2301:| 2280:X 2274:K 2254:, 2251:0 2245:r 2225:) 2222:X 2219:( 2216:C 2196:, 2193:) 2190:X 2187:( 2182:0 2178:C 2167:. 2147:) 2144:X 2141:( 2138:C 2118:, 2115:) 2112:X 2109:( 2100:C 2072:) 2069:X 2066:( 2061:B 2057:C 2032:X 2001:. 1998:X 1978:) 1975:X 1972:( 1967:B 1963:C 1942:) 1939:X 1936:( 1933:C 1913:. 1910:X 1890:) 1887:X 1884:( 1881:C 1853:. 1850:X 1830:) 1827:X 1824:( 1821:C 1795:. 1792:) 1789:X 1786:( 1781:C 1748:X 1728:) 1725:X 1722:( 1717:C 1695:) 1692:X 1689:( 1684:C 1655:) 1652:X 1649:( 1644:C 1615:X 1591:) 1588:X 1585:( 1580:C 1534:. 1530:C 1523:) 1520:X 1517:( 1512:C 1470:. 1467:X 1447:. 1444:) 1441:X 1438:( 1433:C 1380:Y 1360:X 1340:) 1337:Y 1334:( 1331:C 1311:) 1308:X 1305:( 1302:C 1282:. 1279:X 1273:Y 1270:: 1267:f 1247:) 1244:) 1241:y 1238:( 1235:f 1232:( 1229:h 1226:= 1223:) 1220:y 1217:( 1214:) 1211:h 1208:( 1205:F 1185:F 1165:F 1141:) 1138:Y 1135:( 1130:C 1122:) 1119:X 1116:( 1111:C 1106:: 1103:F 1083:Y 1063:X 1052:. 1036:A 1016:. 1013:) 1010:X 1007:( 1002:C 980:A 956:) 953:X 950:( 945:C 919:A 899:. 896:) 893:X 890:( 885:C 868:. 852:, 849:) 846:X 843:( 838:C 808:) 805:X 802:( 797:C 775:K 760:. 737:) 734:X 731:( 726:C 704:X 677:, 674:X 647:) 644:X 641:( 636:C 596:. 593:) 590:X 587:( 557:X 533:. 530:) 527:X 524:( 519:C 498:. 482:X 462:) 459:X 456:( 451:C 427:. 424:) 421:y 418:( 415:f 409:) 406:x 403:( 400:f 380:) 377:X 374:( 369:C 361:f 341:X 335:y 332:, 329:x 309:X 286:) 283:X 280:( 275:C 232:) 229:X 226:( 221:C 199:. 196:X 164:, 160:| 156:) 153:x 150:( 147:f 143:| 137:X 131:x 123:= 117:f 86:, 83:) 80:X 77:( 72:C 42:X

Index

mathematical analysis
functional analysis
continuous functions
compact
Hausdorff space
real
complex numbers
vector space
normed space
uniform norm
topology
uniform convergence
Banach algebra
Rudin 1973
Urysohn's lemma
separates points
locally compact
Riesz–Markov–Kakutani representation theorem
continuous dual space
Radon measures
Borel measures
total variation
ba spaces
Dunford & Schwartz 1958
Positive linear functionals
regular
Borel measures
Rudin 1966
reflexive
weakly

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