1915:
1002:
735:). It is easy to check that the linear form induced by σ is continuous in the sup-norm if σ is bounded, and the result follows since a linear form on the dense subspace of simple functions extends to an element of B(Σ)* if it is continuous in the sup-norm.
867:
1159:
886:
733:
169:
567:
503:
436:
404:
349:
239:
207:
56:
531:
111:
79:
471:
317:
605:
792:
1804:
1640:
1067:
1467:
643:
which allows for a measure to be represented as a linear functional on measurable functions. In particular, this isomorphism allows one to
1630:
1757:
1612:
1588:
1436:
997:{\displaystyle N_{\mu }^{\perp }=\{\sigma \in ba(\Sigma ):\mu (A)=0\Rightarrow \sigma (A)=0{\text{ for any }}A\in \Sigma \},}
1480:
685:
1569:
1460:
1366:
1839:
1484:
655:
additivity). This is due to
Dunford & Schwartz, and is often used to define the integral with respect to
1635:
1918:
1691:
1625:
1453:
640:
1655:
120:
1900:
1854:
1778:
1660:
1895:
1711:
1051:
779:
1747:
1645:
1548:
1939:
1844:
1620:
540:
476:
409:
377:
322:
212:
180:
29:
1944:
1875:
1819:
1783:
1582:
1181:
1016:
636:
516:
96:
64:
1578:
651:
with respect to a finitely additive measure (note that the usual
Lebesgue integral requires
639:
of B(Σ). This is due to
Hildebrandt and Fichtenholtz & Kantorovich. This is a kind of
441:
287:
1858:
1445:
8:
1824:
1762:
1476:
587:
1849:
1716:
1407:
1355:
1271:
1055:
775:
1829:
1432:
1372:
1362:
1008:
273:
114:
1834:
1752:
1721:
1701:
1686:
1681:
1676:
1424:
1397:
1302:
1261:
862:{\displaystyle N_{\mu }:=\{f\in B(\Sigma ):f=0\ \mu {\text{-almost everywhere}}\}.}
1513:
1696:
1650:
1598:
1593:
1564:
679:
510:
352:
86:
1523:
1885:
1737:
1538:
1338:. Mathematical Surveys. Vol. 15. American Mathematical Society. Chapter I.
760:
656:
577:
242:
90:
1933:
1890:
1814:
1543:
1528:
1518:
752:
660:
355:
174:
59:
1428:
1880:
1533:
1503:
1376:
1031:
628:
627:
Let B(Σ) be the space of bounded Σ-measurable functions, equipped with the
371:
82:
1809:
1799:
1706:
1508:
1307:
1290:
1154:{\displaystyle L^{1}(\mu )\subset L^{1}(\mu )^{**}=L^{\infty }(\mu )^{*}}
374:) with respect to the same norm defined by the total variation, and thus
20:
1742:
1574:
1411:
1275:
612:
608:
573:
534:
277:
1402:
1385:
1266:
1249:
764:
648:
616:
250:
1291:"Sur les opérations linéaires dans l'espace des fonctions bornées"
1164:
is isomorphic to the inclusion of the space of countably additive
1229:
1217:
1205:
1168:-a.c. bounded measures inside the space of all finitely additive
1061:-a.c. measures. In other words, the inclusion in the bidual
1475:
1315:
1282:
678:
finitely additive measures σ on Σ and the vector space of
1288:
1070:
889:
795:
688:
590:
543:
519:
479:
444:
412:
380:
325:
290:
215:
183:
123:
99:
67:
32:
1327:
1805:Spectral theory of ordinary differential equations
1418:
1354:
1153:
996:
861:
727:
599:
561:
525:
497:
465:
430:
398:
343:
311:
233:
201:
163:
105:
73:
50:
1419:Kantorovitch, Leonid V.; Akilov, Gleb P. (1982).
1390:Transactions of the American Mathematical Society
1254:Transactions of the American Mathematical Society
1931:
728:{\displaystyle \mu (A)=\zeta \left(1_{A}\right)}
670:(Σ) = B(Σ)* is easy to see. There is an obvious
1321:
1235:
1223:
1211:
1193:
1241:
1461:
1383:
988:
908:
853:
809:
130:
124:
1289:Fichtenholz, G.; Kantorovich, L.V. (1934).
1247:
1468:
1454:
1333:
782:of B(Σ) by the closed subspace of bounded
1401:
1306:
1265:
1758:Group algebra of a locally compact group
370:All three spaces are complete (they are
1352:
1932:
1030:When the measure space is furthermore
1449:
1357:Sequences and series in Banach spaces
1194:Dunford, N.; Schwartz, J.T. (1958).
674:duality between the vector space of
505:for Σ the algebra of Borel sets on
164:{\displaystyle \|\nu \|=|\nu |(X).}
13:
1346:
1250:"On bounded functional operations"
1130:
1054:is identified with the set of all
985:
926:
824:
553:
520:
489:
422:
390:
335:
225:
193:
100:
68:
42:
14:
1956:
1914:
1913:
1840:Topological quantum field theory
738:
584:(2), is often denoted as simply
276:, and Σ is the sigma-algebra of
1384:Yosida, K.; Hewitt, E. (1952).
1334:Diestel, J.; Uhl, J.J. (1977).
763:positive measure on Σ then the
659:, and especially vector-valued
622:
1142:
1135:
1110:
1103:
1087:
1081:
965:
959:
953:
944:
938:
929:
923:
827:
821:
698:
692:
556:
550:
492:
486:
460:
454:
425:
419:
393:
387:
338:
332:
306:
300:
228:
222:
196:
190:
155:
149:
145:
137:
45:
39:
1:
1636:Uniform boundedness principle
1187:
365:
113:. The norm is defined as the
1386:"Finitely additive measures"
641:Riesz representation theorem
209:is defined as the subset of
7:
1322:Dunford & Schwartz 1958
1236:Dunford & Schwartz 1958
1224:Dunford & Schwartz 1958
1212:Dunford & Schwartz 1958
1175:
562:{\displaystyle ba(\Sigma )}
498:{\displaystyle ca(\Sigma )}
431:{\displaystyle ba(\Sigma )}
399:{\displaystyle ca(\Sigma )}
344:{\displaystyle ca(\Sigma )}
243:countably additive measures
234:{\displaystyle ba(\Sigma )}
202:{\displaystyle ca(\Sigma )}
51:{\displaystyle ba(\Sigma )}
10:
1961:
1779:Invariant subspace problem
1248:Hildebrandt, T.H. (1934).
778:norm is by definition the
1909:
1868:
1792:
1771:
1730:
1669:
1611:
1557:
1499:
1492:
880:)* is thus isomorphic to
1748:Spectrum of a C*-algebra
1353:Diestel, Joseph (1984).
1196:Linear operators, Part I
1172:-a.c. bounded measures.
666:The topological duality
1845:Noncommutative geometry
1429:10.1016/C2013-0-03044-7
526:{\displaystyle \Sigma }
106:{\displaystyle \Sigma }
74:{\displaystyle \Sigma }
1901:Tomita–Takesaki theory
1876:Approximation property
1820:Calculus of variations
1155:
998:
872:The dual Banach space
863:
729:
601:
563:
527:
499:
467:
466:{\displaystyle rca(X)}
432:
406:is a closed subset of
400:
345:
313:
312:{\displaystyle rca(X)}
235:
203:
165:
107:
89:and finitely additive
75:
52:
16:Class of Banach spaces
1896:Banach–Mazur distance
1859:Generalized functions
1198:. Wiley-Interscience.
1182:List of Banach spaces
1156:
1052:Radon–Nikodym theorem
1042:) is in turn dual to
1017:absolutely continuous
999:
864:
730:
637:continuous dual space
602:
564:
528:
500:
468:
433:
401:
346:
314:
236:
204:
166:
108:
76:
53:
1641:Kakutani fixed-point
1626:Riesz representation
1308:10.4064/sm-5-1-69-98
1068:
887:
793:
686:
588:
572:The ba space of the
541:
517:
477:
442:
410:
378:
323:
288:
213:
181:
121:
97:
65:
30:
1825:Functional calculus
1784:Mahler's conjecture
1763:Von Neumann algebra
1477:Functional analysis
1421:Functional Analysis
1361:. Springer-Verlag.
1011:signed measures on
976: for any
904:
774:) endowed with the
635:(Σ) = B(Σ)* is the
473:is a closed set of
319:is the subspace of
1850:Riemann hypothesis
1549:Topological vector
1295:Studia Mathematica
1151:
1056:countably additive
1027:-a.c. for short).
1007:i.e. the space of
994:
890:
859:
850:-almost everywhere
776:essential supremum
725:
600:{\displaystyle ba}
597:
559:
523:
495:
463:
428:
396:
351:consisting of all
341:
309:
263:countably additive
231:
199:
161:
103:
85:consisting of all
71:
48:
1927:
1926:
1830:Integral operator
1607:
1606:
1438:978-0-08-023036-8
1009:finitely additive
977:
851:
844:
786:-null functions:
274:topological space
177:, then the space
1952:
1917:
1916:
1835:Jones polynomial
1753:Operator algebra
1497:
1496:
1470:
1463:
1456:
1447:
1446:
1442:
1415:
1405:
1380:
1360:
1340:
1339:
1331:
1325:
1319:
1313:
1312:
1310:
1286:
1280:
1279:
1269:
1245:
1239:
1233:
1227:
1221:
1215:
1209:
1199:
1160:
1158:
1157:
1152:
1150:
1149:
1134:
1133:
1121:
1120:
1102:
1101:
1080:
1079:
1050:), which by the
1019:with respect to
1003:
1001:
1000:
995:
978:
975:
903:
898:
868:
866:
865:
860:
852:
849:
842:
805:
804:
734:
732:
731:
726:
724:
720:
719:
680:simple functions
606:
604:
603:
598:
568:
566:
565:
560:
532:
530:
529:
524:
511:simple functions
509:. The space of
504:
502:
501:
496:
472:
470:
469:
464:
437:
435:
434:
429:
405:
403:
402:
397:
350:
348:
347:
342:
318:
316:
315:
310:
255:bounded additive
240:
238:
237:
232:
208:
206:
205:
200:
170:
168:
167:
162:
148:
140:
112:
110:
109:
104:
80:
78:
77:
72:
57:
55:
54:
49:
1960:
1959:
1955:
1954:
1953:
1951:
1950:
1949:
1930:
1929:
1928:
1923:
1905:
1869:Advanced topics
1864:
1788:
1767:
1726:
1692:Hilbert–Schmidt
1665:
1656:Gelfand–Naimark
1603:
1553:
1488:
1474:
1439:
1403:10.2307/1990654
1369:
1349:
1347:Further reading
1344:
1343:
1336:Vector measures
1332:
1328:
1320:
1316:
1287:
1283:
1267:10.2307/1989829
1246:
1242:
1234:
1230:
1222:
1218:
1210:
1206:
1190:
1178:
1145:
1141:
1129:
1125:
1113:
1109:
1097:
1093:
1075:
1071:
1069:
1066:
1065:
974:
899:
894:
888:
885:
884:
848:
800:
796:
794:
791:
790:
749:
715:
711:
707:
687:
684:
683:
657:vector measures
625:
589:
586:
585:
578:natural numbers
542:
539:
538:
518:
515:
514:
478:
475:
474:
443:
440:
439:
411:
408:
407:
379:
376:
375:
368:
324:
321:
320:
289:
286:
285:
245:. The notation
214:
211:
210:
182:
179:
178:
144:
136:
122:
119:
118:
98:
95:
94:
91:signed measures
66:
63:
62:
60:algebra of sets
31:
28:
27:
17:
12:
11:
5:
1958:
1948:
1947:
1942:
1940:Measure theory
1925:
1924:
1922:
1921:
1910:
1907:
1906:
1904:
1903:
1898:
1893:
1888:
1886:Choquet theory
1883:
1878:
1872:
1870:
1866:
1865:
1863:
1862:
1852:
1847:
1842:
1837:
1832:
1827:
1822:
1817:
1812:
1807:
1802:
1796:
1794:
1790:
1789:
1787:
1786:
1781:
1775:
1773:
1769:
1768:
1766:
1765:
1760:
1755:
1750:
1745:
1740:
1738:Banach algebra
1734:
1732:
1728:
1727:
1725:
1724:
1719:
1714:
1709:
1704:
1699:
1694:
1689:
1684:
1679:
1673:
1671:
1667:
1666:
1664:
1663:
1661:Banach–Alaoglu
1658:
1653:
1648:
1643:
1638:
1633:
1628:
1623:
1617:
1615:
1609:
1608:
1605:
1604:
1602:
1601:
1596:
1591:
1589:Locally convex
1586:
1572:
1567:
1561:
1559:
1555:
1554:
1552:
1551:
1546:
1541:
1536:
1531:
1526:
1521:
1516:
1511:
1506:
1500:
1494:
1490:
1489:
1473:
1472:
1465:
1458:
1450:
1444:
1443:
1437:
1416:
1381:
1367:
1348:
1345:
1342:
1341:
1326:
1314:
1281:
1260:(4): 868–875.
1240:
1228:
1216:
1203:
1202:
1201:
1200:
1189:
1186:
1185:
1184:
1177:
1174:
1162:
1161:
1148:
1144:
1140:
1137:
1132:
1128:
1124:
1119:
1116:
1112:
1108:
1105:
1100:
1096:
1092:
1089:
1086:
1083:
1078:
1074:
1005:
1004:
993:
990:
987:
984:
981:
973:
970:
967:
964:
961:
958:
955:
952:
949:
946:
943:
940:
937:
934:
931:
928:
925:
922:
919:
916:
913:
910:
907:
902:
897:
893:
870:
869:
858:
855:
847:
841:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
808:
803:
799:
780:quotient space
761:sigma-additive
748:
737:
723:
718:
714:
710:
706:
703:
700:
697:
694:
691:
661:Radon measures
624:
621:
596:
593:
558:
555:
552:
549:
546:
522:
494:
491:
488:
485:
482:
462:
459:
456:
453:
450:
447:
427:
424:
421:
418:
415:
395:
392:
389:
386:
383:
367:
364:
356:Borel measures
340:
337:
334:
331:
328:
308:
305:
302:
299:
296:
293:
241:consisting of
230:
227:
224:
221:
218:
198:
195:
192:
189:
186:
160:
157:
154:
151:
147:
143:
139:
135:
132:
129:
126:
102:
70:
47:
44:
41:
38:
35:
15:
9:
6:
4:
3:
2:
1957:
1946:
1945:Banach spaces
1943:
1941:
1938:
1937:
1935:
1920:
1912:
1911:
1908:
1902:
1899:
1897:
1894:
1892:
1891:Weak topology
1889:
1887:
1884:
1882:
1879:
1877:
1874:
1873:
1871:
1867:
1860:
1856:
1853:
1851:
1848:
1846:
1843:
1841:
1838:
1836:
1833:
1831:
1828:
1826:
1823:
1821:
1818:
1816:
1815:Index theorem
1813:
1811:
1808:
1806:
1803:
1801:
1798:
1797:
1795:
1791:
1785:
1782:
1780:
1777:
1776:
1774:
1772:Open problems
1770:
1764:
1761:
1759:
1756:
1754:
1751:
1749:
1746:
1744:
1741:
1739:
1736:
1735:
1733:
1729:
1723:
1720:
1718:
1715:
1713:
1710:
1708:
1705:
1703:
1700:
1698:
1695:
1693:
1690:
1688:
1685:
1683:
1680:
1678:
1675:
1674:
1672:
1668:
1662:
1659:
1657:
1654:
1652:
1649:
1647:
1644:
1642:
1639:
1637:
1634:
1632:
1629:
1627:
1624:
1622:
1619:
1618:
1616:
1614:
1610:
1600:
1597:
1595:
1592:
1590:
1587:
1584:
1580:
1576:
1573:
1571:
1568:
1566:
1563:
1562:
1560:
1556:
1550:
1547:
1545:
1542:
1540:
1537:
1535:
1532:
1530:
1527:
1525:
1522:
1520:
1517:
1515:
1512:
1510:
1507:
1505:
1502:
1501:
1498:
1495:
1491:
1486:
1482:
1478:
1471:
1466:
1464:
1459:
1457:
1452:
1451:
1448:
1440:
1434:
1430:
1426:
1422:
1417:
1413:
1409:
1404:
1399:
1395:
1391:
1387:
1382:
1378:
1374:
1370:
1368:0-387-90859-5
1364:
1359:
1358:
1351:
1350:
1337:
1330:
1323:
1318:
1309:
1304:
1300:
1296:
1292:
1285:
1277:
1273:
1268:
1263:
1259:
1255:
1251:
1244:
1237:
1232:
1225:
1220:
1213:
1208:
1204:
1197:
1192:
1191:
1183:
1180:
1179:
1173:
1171:
1167:
1146:
1138:
1126:
1122:
1117:
1114:
1106:
1098:
1094:
1090:
1084:
1076:
1072:
1064:
1063:
1062:
1060:
1057:
1053:
1049:
1045:
1041:
1037:
1033:
1028:
1026:
1022:
1018:
1014:
1010:
991:
982:
979:
971:
968:
962:
956:
950:
947:
941:
935:
932:
920:
917:
914:
911:
905:
900:
895:
891:
883:
882:
881:
879:
875:
856:
845:
839:
836:
833:
830:
818:
815:
812:
806:
801:
797:
789:
788:
787:
785:
781:
777:
773:
769:
766:
762:
758:
754:
753:sigma-algebra
746:
742:
736:
721:
716:
712:
708:
704:
701:
695:
689:
681:
677:
673:
669:
664:
662:
658:
654:
650:
646:
642:
638:
634:
630:
620:
618:
614:
610:
594:
591:
583:
579:
575:
570:
547:
544:
536:
512:
508:
483:
480:
457:
451:
448:
445:
416:
413:
384:
381:
373:
372:Banach spaces
363:
361:
357:
354:
329:
326:
303:
297:
294:
291:
283:
279:
275:
271:
266:
264:
261:is short for
260:
256:
252:
248:
244:
219:
216:
187:
184:
176:
175:sigma-algebra
171:
158:
152:
141:
133:
127:
116:
92:
88:
84:
61:
36:
33:
26:
22:
1881:Balanced set
1855:Distribution
1793:Applications
1646:Krein–Milman
1631:Closed graph
1423:. Pergamon.
1420:
1396:(1): 46–66.
1393:
1389:
1356:
1335:
1329:
1317:
1298:
1294:
1284:
1257:
1253:
1243:
1231:
1219:
1207:
1195:
1169:
1165:
1163:
1058:
1047:
1043:
1039:
1035:
1032:sigma-finite
1029:
1024:
1020:
1012:
1006:
877:
873:
871:
783:
771:
767:
756:
750:
744:
740:
675:
671:
667:
665:
652:
644:
632:
629:uniform norm
626:
623:Dual of B(Σ)
581:
571:
506:
369:
359:
281:
269:
267:
262:
258:
254:
246:
172:
83:Banach space
24:
18:
1810:Heat kernel
1800:Hardy space
1707:Trace class
1621:Hahn–Banach
1583:Topological
21:mathematics
1934:Categories
1743:C*-algebra
1558:Properties
1238:, IV.2.17.
1226:, IV.2.16.
1214:, IV.2.15.
1188:References
751:If Σ is a
613:dual space
609:isomorphic
366:Properties
278:Borel sets
173:If Σ is a
117:, that is
1717:Unbounded
1712:Transpose
1670:Operators
1599:Separable
1594:Reflexive
1579:Algebraic
1565:Barrelled
1301:: 69–98.
1147:∗
1139:μ
1131:∞
1118:∗
1115:∗
1107:μ
1091:⊂
1085:μ
1015:that are
986:Σ
983:∈
957:σ
954:⇒
936:μ
927:Σ
915:∈
912:σ
901:⊥
896:μ
846:μ
825:Σ
816:∈
802:μ
705:ζ
690:μ
672:algebraic
653:countable
574:power set
554:Σ
521:Σ
490:Σ
423:Σ
391:Σ
336:Σ
226:Σ
194:Σ
142:ν
131:‖
128:ν
125:‖
115:variation
101:Σ
69:Σ
43:Σ
1919:Category
1731:Algebras
1613:Theorems
1570:Complete
1539:Schwartz
1485:glossary
1176:See also
765:Lp space
739:Dual of
649:integral
631:. Then
251:mnemonic
25:ba space
1722:Unitary
1702:Nuclear
1687:Compact
1682:Bounded
1677:Adjoint
1651:Min–max
1544:Sobolev
1529:Nuclear
1519:Hilbert
1514:Fréchet
1479: (
1412:1990654
1377:9556781
1276:1989829
617:ℓ space
615:of the
611:to the
607:and is
576:of the
353:regular
284:, then
87:bounded
81:is the
1697:Normal
1534:Orlicz
1524:Hölder
1504:Banach
1493:Spaces
1481:topics
1435:
1410:
1375:
1365:
1274:
843:
645:define
438:, and
58:of an
23:, the
1509:Besov
1408:JSTOR
1272:JSTOR
1034:then
759:is a
535:dense
272:is a
249:is a
1857:(or
1575:Dual
1433:ISBN
1373:OCLC
1363:ISBN
755:and
647:the
257:and
253:for
1425:doi
1398:doi
1303:doi
1262:doi
676:all
537:in
533:is
513:on
358:on
280:in
268:If
93:on
19:In
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1406:.
1394:72
1392:.
1388:.
1371:.
1297:.
1293:.
1270:.
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1256:.
1252:.
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668:ba
663:.
633:ba
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580:,
569:.
362:.
265:.
259:ca
247:ba
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1581:/
1577:(
1487:)
1469:e
1462:t
1455:v
1441:.
1427::
1414:.
1400::
1379:.
1324:.
1311:.
1305::
1299:5
1278:.
1264::
1170:μ
1166:μ
1143:)
1136:(
1127:L
1123:=
1111:)
1104:(
1099:1
1095:L
1088:)
1082:(
1077:1
1073:L
1059:μ
1048:μ
1046:(
1044:L
1040:μ
1038:(
1036:L
1025:μ
1023:(
1021:μ
1013:Σ
992:,
989:}
980:A
972:0
969:=
966:)
963:A
960:(
951:0
948:=
945:)
942:A
939:(
933::
930:)
924:(
921:a
918:b
909:{
906:=
892:N
878:μ
876:(
874:L
857:.
854:}
840:0
837:=
834:f
831::
828:)
822:(
819:B
813:f
810:{
798:N
784:μ
772:μ
770:(
768:L
757:μ
747:)
745:μ
743:(
741:L
722:)
717:A
713:1
709:(
702:=
699:)
696:A
693:(
682:(
595:a
592:b
557:)
551:(
548:a
545:b
507:X
493:)
487:(
484:a
481:c
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458:X
455:(
452:a
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446:r
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420:(
417:a
414:b
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385:a
382:c
360:X
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333:(
330:a
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301:(
298:a
295:c
292:r
282:X
270:X
229:)
223:(
220:a
217:b
197:)
191:(
188:a
185:c
159:.
156:)
153:X
150:(
146:|
138:|
134:=
46:)
40:(
37:a
34:b
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