1478:
1514:
259:
192:
941:
873:
133:
1367:
802:
706:
595:
568:
512:
481:
433:
389:
352:
319:
534:
1507:
893:
825:
770:
750:
726:
670:
650:
630:
1030:
1500:
1193:
1783:
1320:
1175:
1151:
1725:
1552:
1043:
1674:
1132:
1023:
1000:
978:
970:
1628:
1598:
1523:
1402:
992:
1047:
1613:
1198:
1254:
1481:
1203:
1188:
1016:
805:
1218:
1746:
1463:
1223:
1741:
1567:
1417:
1341:
1458:
952:
48:
1274:
1208:
1669:
1562:
1310:
1111:
205:
138:
1183:
1542:
1407:
898:
830:
1438:
1382:
1346:
1694:
1537:
1421:
605:
94:
1654:
1387:
1325:
1039:
1412:
1279:
775:
679:
573:
541:
490:
454:
47:
Numerous well-known results may be derived from the
Freudenthal spectral theorem. The well-known
1608:
405:
361:
324:
291:
1720:
1638:
1392:
996:
974:
598:
519:
1664:
1623:
1593:
1588:
1397:
1315:
1284:
1264:
1249:
1244:
1239:
1076:
56:
29:
1704:
1259:
1213:
1161:
1156:
1127:
1008:
878:
810:
755:
735:
711:
673:
655:
635:
615:
60:
41:
1086:
875:
function and establishes an isometric lattice isomorphism between the band generated by
1684:
1679:
1659:
1633:
1618:
1603:
1572:
1448:
1300:
1101:
708:(in the usual sense). Moreover, since by the Freudenthal spectral theorem, any measure
601:
515:
52:
1777:
1453:
1377:
1106:
1091:
1081:
484:
1492:
1443:
1096:
1066:
32:
in 1936. It roughly states that any element dominated by a positive element in a
63:
can all be shown to follow as special cases of the
Freudenthal spectral theorem.
1547:
1372:
1362:
1269:
1071:
729:
609:
392:
195:
37:
33:
25:
17:
1689:
1557:
1305:
1145:
1141:
1137:
1699:
652:-simple functions (as defined above) can be shown to correspond exactly to
1496:
1012:
272:
be any Riesz space with the principal projection property and
901:
881:
833:
813:
778:
758:
738:
714:
682:
658:
638:
618:
576:
544:
522:
493:
457:
408:
364:
327:
294:
208:
141:
97:
1755:
1734:
1713:
1647:
1581:
1530:
1431:
1355:
1334:
1293:
1232:
1174:
1120:
1055:
1368:Spectral theory of ordinary differential equations
935:
887:
867:
819:
796:
764:
744:
720:
700:
664:
644:
624:
589:
562:
528:
506:
475:
427:
383:
346:
313:
253:
186:
127:
987:Zaanen, Adriaan C.; Luxemburg, W. A. J. (1971),
967:Introduction to Operator Theory in Riesz spaces
752:can be monotonously approximated from below by
1508:
1024:
268:The Freudenthal spectral theorem states: Let
8:
422:
409:
378:
365:
341:
328:
308:
295:
40:can in a sense be approximated uniformly by
1515:
1501:
1493:
1059:
1031:
1017:
1009:
906:
900:
880:
838:
832:
812:
777:
757:
737:
713:
681:
657:
637:
617:
581:
575:
543:
521:
498:
492:
456:
416:
407:
372:
363:
335:
326:
302:
293:
254:{\displaystyle p_{1},p_{2},\ldots ,p_{n}}
245:
226:
213:
207:
187:{\displaystyle p_{1},p_{2},\ldots ,p_{n}}
178:
159:
146:
140:
96:
75:be any positive element in a Riesz space
1321:Group algebra of a locally compact group
806:Lebesgue's monotone convergence theorem
936:{\displaystyle L^{1}(X,\Sigma ,\mu )}
868:{\displaystyle L^{1}(X,\Sigma ,\mu )}
447:Relation to the Radon–Nikodym theorem
435:is monotone decreasing and converges
391:is monotone increasing and converges
7:
284:in the principal ideal generated by
921:
853:
788:
692:
554:
467:
14:
1524:Ordered topological vector spaces
827:can be shown to correspond to an
202:, any real linear combination of
1477:
1476:
1403:Topological quantum field theory
772:-measurable simple functions on
1784:Theorems in functional analysis
128:{\displaystyle p\wedge (e-p)=0}
930:
912:
862:
844:
791:
779:
695:
683:
557:
545:
470:
458:
358:-simple functions, such that
116:
104:
1:
1614:Locally convex vector lattice
1199:Uniform boundedness principle
610:principal projection property
38:principal projection property
22:Freudenthal spectral theorem
965:Zaanen, Adriaan C. (1996),
797:{\displaystyle (X,\Sigma )}
701:{\displaystyle (X,\Sigma )}
612:. For any positive measure
590:{\displaystyle M_{\sigma }}
563:{\displaystyle (X,\Sigma )}
507:{\displaystyle M_{\sigma }}
476:{\displaystyle (X,\Sigma )}
1800:
1568:Topological vector lattice
1342:Invariant subspace problem
1648:Types of elements/subsets
1472:
1062:
428:{\displaystyle \{t_{n}\}}
384:{\displaystyle \{s_{n}\}}
347:{\displaystyle \{t_{n}\}}
314:{\displaystyle \{s_{n}\}}
87:is called a component of
1563:Positive linear operator
1311:Spectrum of a C*-algebra
288:, there exist sequences
276:any positive element in
79:. A positive element of
1543:Partially ordered space
1408:Noncommutative geometry
895:and the Banach Lattice
570:. It can be shown that
529:{\displaystyle \sigma }
280:. Then for any element
1714:Topologies/Convergence
1582:Types of orders/spaces
1464:Tomita–Takesaki theory
1439:Approximation property
1383:Calculus of variations
937:
889:
869:
821:
798:
766:
746:
722:
702:
666:
646:
626:
591:
564:
530:
508:
477:
429:
385:
348:
315:
255:
188:
129:
51:, the validity of the
1459:Banach–Mazur distance
1422:Generalized functions
953:Radon–Nikodym theorem
938:
890:
870:
822:
799:
767:
747:
723:
703:
667:
647:
627:
592:
565:
531:
509:
478:
430:
386:
349:
316:
256:
189:
130:
49:Radon–Nikodym theorem
1763:Freudenthal spectral
1695:Quasi-interior point
1538:Ordered vector space
1204:Kakutani fixed-point
1189:Riesz representation
899:
888:{\displaystyle \mu }
879:
831:
820:{\displaystyle \nu }
811:
776:
765:{\displaystyle \mu }
756:
745:{\displaystyle \mu }
736:
721:{\displaystyle \nu }
712:
680:
665:{\displaystyle \mu }
656:
645:{\displaystyle \mu }
636:
625:{\displaystyle \mu }
616:
608:, and hence has the
606:total variation norm
574:
542:
520:
491:
455:
406:
362:
325:
292:
206:
139:
95:
1388:Functional calculus
1347:Mahler's conjecture
1326:Von Neumann algebra
1040:Functional analysis
59:from the theory of
1413:Riemann hypothesis
1112:Topological vector
933:
885:
865:
817:
794:
762:
742:
718:
698:
662:
642:
622:
587:
560:
536:-additive measures
526:
514:the real space of
504:
473:
425:
381:
344:
311:
265:-simple function.
251:
184:
125:
26:Riesz space theory
1771:
1770:
1721:Order convergence
1639:Regularly ordered
1490:
1489:
1393:Integral operator
1170:
1169:
599:Dedekind complete
1791:
1665:Lattice disjoint
1624:Order bound dual
1517:
1510:
1503:
1494:
1480:
1479:
1398:Jones polynomial
1316:Operator algebra
1060:
1033:
1026:
1019:
1010:
1005:
983:
942:
940:
939:
934:
911:
910:
894:
892:
891:
886:
874:
872:
871:
866:
843:
842:
826:
824:
823:
818:
803:
801:
800:
795:
771:
769:
768:
763:
751:
749:
748:
743:
727:
725:
724:
719:
707:
705:
704:
699:
674:simple functions
671:
669:
668:
663:
651:
649:
648:
643:
631:
629:
628:
623:
596:
594:
593:
588:
586:
585:
569:
567:
566:
561:
535:
533:
532:
527:
513:
511:
510:
505:
503:
502:
482:
480:
479:
474:
434:
432:
431:
426:
421:
420:
390:
388:
387:
382:
377:
376:
353:
351:
350:
345:
340:
339:
320:
318:
317:
312:
307:
306:
260:
258:
257:
252:
250:
249:
231:
230:
218:
217:
193:
191:
190:
185:
183:
182:
164:
163:
151:
150:
134:
132:
131:
126:
61:normal operators
57:spectral theorem
42:simple functions
30:Hans Freudenthal
1799:
1798:
1794:
1793:
1792:
1790:
1789:
1788:
1774:
1773:
1772:
1767:
1751:
1730:
1709:
1705:Weak order unit
1670:Dual/Polar cone
1643:
1609:Fréchet lattice
1577:
1526:
1521:
1491:
1486:
1468:
1432:Advanced topics
1427:
1351:
1330:
1289:
1255:Hilbert–Schmidt
1228:
1219:Gelfand–Naimark
1166:
1116:
1051:
1037:
1003:
986:
981:
964:
961:
949:
902:
897:
896:
877:
876:
834:
829:
828:
809:
808:
774:
773:
754:
753:
734:
733:
710:
709:
678:
677:
654:
653:
634:
633:
614:
613:
577:
572:
571:
540:
539:
518:
517:
494:
489:
488:
453:
452:
449:
412:
404:
403:
368:
360:
359:
331:
323:
322:
298:
290:
289:
241:
222:
209:
204:
203:
174:
155:
142:
137:
136:
93:
92:
69:
53:Poisson formula
24:is a result in
12:
11:
5:
1797:
1795:
1787:
1786:
1776:
1775:
1769:
1768:
1766:
1765:
1759:
1757:
1753:
1752:
1750:
1749:
1744:
1738:
1736:
1732:
1731:
1729:
1728:
1726:Order topology
1723:
1717:
1715:
1711:
1710:
1708:
1707:
1702:
1697:
1692:
1687:
1685:Order summable
1682:
1680:Order complete
1677:
1672:
1667:
1662:
1660:Cone-saturated
1657:
1651:
1649:
1645:
1644:
1642:
1641:
1636:
1634:Order complete
1631:
1626:
1621:
1619:Normed lattice
1616:
1611:
1606:
1604:Banach lattice
1601:
1596:
1591:
1585:
1583:
1579:
1578:
1576:
1575:
1573:Vector lattice
1570:
1565:
1560:
1555:
1553:Order topology
1550:
1545:
1540:
1534:
1532:
1531:Basic concepts
1528:
1527:
1522:
1520:
1519:
1512:
1505:
1497:
1488:
1487:
1485:
1484:
1473:
1470:
1469:
1467:
1466:
1461:
1456:
1451:
1449:Choquet theory
1446:
1441:
1435:
1433:
1429:
1428:
1426:
1425:
1415:
1410:
1405:
1400:
1395:
1390:
1385:
1380:
1375:
1370:
1365:
1359:
1357:
1353:
1352:
1350:
1349:
1344:
1338:
1336:
1332:
1331:
1329:
1328:
1323:
1318:
1313:
1308:
1303:
1301:Banach algebra
1297:
1295:
1291:
1290:
1288:
1287:
1282:
1277:
1272:
1267:
1262:
1257:
1252:
1247:
1242:
1236:
1234:
1230:
1229:
1227:
1226:
1224:Banach–Alaoglu
1221:
1216:
1211:
1206:
1201:
1196:
1191:
1186:
1180:
1178:
1172:
1171:
1168:
1167:
1165:
1164:
1159:
1154:
1152:Locally convex
1149:
1135:
1130:
1124:
1122:
1118:
1117:
1115:
1114:
1109:
1104:
1099:
1094:
1089:
1084:
1079:
1074:
1069:
1063:
1057:
1053:
1052:
1038:
1036:
1035:
1028:
1021:
1013:
1007:
1006:
1001:
989:Riesz spaces I
984:
979:
960:
957:
956:
955:
948:
945:
932:
929:
926:
923:
920:
917:
914:
909:
905:
884:
864:
861:
858:
855:
852:
849:
846:
841:
837:
816:
793:
790:
787:
784:
781:
761:
741:
730:band generated
717:
697:
694:
691:
688:
685:
661:
641:
621:
602:Banach Lattice
584:
580:
559:
556:
553:
550:
547:
525:
501:
497:
472:
469:
466:
463:
460:
448:
445:
439:-uniformly to
424:
419:
415:
411:
380:
375:
371:
367:
343:
338:
334:
330:
310:
305:
301:
297:
248:
244:
240:
237:
234:
229:
225:
221:
216:
212:
198:components of
181:
177:
173:
170:
167:
162:
158:
154:
149:
145:
124:
121:
118:
115:
112:
109:
106:
103:
100:
68:
65:
13:
10:
9:
6:
4:
3:
2:
1796:
1785:
1782:
1781:
1779:
1764:
1761:
1760:
1758:
1754:
1748:
1745:
1743:
1740:
1739:
1737:
1733:
1727:
1724:
1722:
1719:
1718:
1716:
1712:
1706:
1703:
1701:
1698:
1696:
1693:
1691:
1688:
1686:
1683:
1681:
1678:
1676:
1673:
1671:
1668:
1666:
1663:
1661:
1658:
1656:
1653:
1652:
1650:
1646:
1640:
1637:
1635:
1632:
1630:
1627:
1625:
1622:
1620:
1617:
1615:
1612:
1610:
1607:
1605:
1602:
1600:
1597:
1595:
1592:
1590:
1587:
1586:
1584:
1580:
1574:
1571:
1569:
1566:
1564:
1561:
1559:
1556:
1554:
1551:
1549:
1546:
1544:
1541:
1539:
1536:
1535:
1533:
1529:
1525:
1518:
1513:
1511:
1506:
1504:
1499:
1498:
1495:
1483:
1475:
1474:
1471:
1465:
1462:
1460:
1457:
1455:
1454:Weak topology
1452:
1450:
1447:
1445:
1442:
1440:
1437:
1436:
1434:
1430:
1423:
1419:
1416:
1414:
1411:
1409:
1406:
1404:
1401:
1399:
1396:
1394:
1391:
1389:
1386:
1384:
1381:
1379:
1378:Index theorem
1376:
1374:
1371:
1369:
1366:
1364:
1361:
1360:
1358:
1354:
1348:
1345:
1343:
1340:
1339:
1337:
1335:Open problems
1333:
1327:
1324:
1322:
1319:
1317:
1314:
1312:
1309:
1307:
1304:
1302:
1299:
1298:
1296:
1292:
1286:
1283:
1281:
1278:
1276:
1273:
1271:
1268:
1266:
1263:
1261:
1258:
1256:
1253:
1251:
1248:
1246:
1243:
1241:
1238:
1237:
1235:
1231:
1225:
1222:
1220:
1217:
1215:
1212:
1210:
1207:
1205:
1202:
1200:
1197:
1195:
1192:
1190:
1187:
1185:
1182:
1181:
1179:
1177:
1173:
1163:
1160:
1158:
1155:
1153:
1150:
1147:
1143:
1139:
1136:
1134:
1131:
1129:
1126:
1125:
1123:
1119:
1113:
1110:
1108:
1105:
1103:
1100:
1098:
1095:
1093:
1090:
1088:
1085:
1083:
1080:
1078:
1075:
1073:
1070:
1068:
1065:
1064:
1061:
1058:
1054:
1049:
1045:
1041:
1034:
1029:
1027:
1022:
1020:
1015:
1014:
1011:
1004:
1002:0-7204-2451-8
998:
994:
993:North-Holland
990:
985:
982:
980:3-540-61989-5
976:
972:
968:
963:
962:
958:
954:
951:
950:
946:
944:
927:
924:
918:
915:
907:
903:
882:
859:
856:
850:
847:
839:
835:
814:
807:
785:
782:
759:
739:
731:
715:
689:
686:
675:
659:
639:
619:
611:
607:
603:
600:
582:
578:
551:
548:
537:
523:
499:
495:
486:
485:measure space
464:
461:
446:
444:
442:
438:
417:
413:
401:
397:
395:
373:
369:
357:
336:
332:
303:
299:
287:
283:
279:
275:
271:
266:
264:
261:is called an
246:
242:
238:
235:
232:
227:
223:
219:
214:
210:
201:
197:
194:are pairwise
179:
175:
171:
168:
165:
160:
156:
152:
147:
143:
122:
119:
113:
110:
107:
101:
98:
90:
86:
82:
78:
74:
66:
64:
62:
58:
54:
50:
45:
43:
39:
35:
31:
27:
23:
19:
1762:
1756:Main results
1444:Balanced set
1418:Distribution
1356:Applications
1209:Krein–Milman
1194:Closed graph
988:
966:
672:-measurable
450:
440:
436:
399:
393:
355:
285:
281:
277:
273:
269:
267:
262:
199:
88:
84:
80:
76:
72:
70:
46:
21:
15:
1675:Normal cone
1599:Archimedean
1548:Riesz space
1373:Heat kernel
1363:Hardy space
1270:Trace class
1184:Hahn–Banach
1146:Topological
34:Riesz space
18:mathematics
1690:Order unit
1629:Order dual
1558:Order unit
1306:C*-algebra
1121:Properties
959:References
396:-uniformly
28:proved by
1735:Operators
1700:Solid set
1280:Unbounded
1275:Transpose
1233:Operators
1162:Separable
1157:Reflexive
1142:Algebraic
1128:Barrelled
928:μ
922:Σ
883:μ
860:μ
854:Σ
815:ν
789:Σ
760:μ
740:μ
716:ν
693:Σ
660:μ
640:μ
620:μ
604:with the
583:σ
555:Σ
524:σ
500:σ
468:Σ
236:…
169:…
111:−
102:∧
67:Statement
36:with the
1778:Category
1742:Positive
1594:AM-space
1589:AL-space
1482:Category
1294:Algebras
1176:Theorems
1133:Complete
1102:Schwartz
1048:glossary
971:Springer
947:See also
196:disjoint
55:and the
1285:Unitary
1265:Nuclear
1250:Compact
1245:Bounded
1240:Adjoint
1214:Min–max
1107:Sobolev
1092:Nuclear
1082:Hilbert
1077:Fréchet
1042: (
728:in the
516:signed
402:, and
1260:Normal
1097:Orlicz
1087:Hölder
1067:Banach
1056:Spaces
1044:topics
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804:, by
597:is a
483:be a
135:. If
1655:Band
1420:(or
1138:Dual
997:ISBN
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487:and
451:Let
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676:on
538:on
398:to
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374:n
370:s
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211:p
200:e
180:n
176:p
172:,
166:,
161:2
157:p
153:,
148:1
144:p
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117:)
114:p
108:e
105:(
99:p
89:e
85:E
81:p
77:E
73:e
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