2062:
1214:
781:
243:
926:
119:
1281:
1091:
1041:
429:
1001:
166:
486:
1116:
960:
1951:
1144:
546:
506:
635:
1475:
Deutsch, Frank; Kenderov, Petar (January 1983). "Continuous
Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections".
1614:
1236:
829:
717:
697:
677:
657:
566:
526:
1150:
1777:
2086:
1904:
1759:
1735:
1295:
431:
a multifunction all of whose values are compact and convex. If graph(Ί) is closed, then for every Δ > 0 there exists a continuous function
1451:
271:
implies that a selection function always exists; however, it is often important that the selection have some "nice" properties, such as
1627:
1716:
1607:
1366:
1535:
Yannelis, Nicholas C.; Prabhakar, N. D. (1983-12-01). "Existence of maximal elements and equilibria in linear topological spaces".
1986:
1320:
1315:
1631:
1305:
723:
1782:
182:
1838:
2065:
1787:
1772:
1600:
1310:
1802:
79:
2047:
1807:
1300:
1246:
2001:
1925:
1332:
1050:
2042:
579:, whose conditions are more general than those of Michael's theorem (and thus the selection is only approximate):
1858:
297:
1792:
1894:
1695:
867:
832:
1767:
1010:
575:
Another set of sufficient conditions for the existence of a continuous approximate selection is given by the
1991:
2022:
1966:
1930:
1544:
41:
33:
1387:
401:
972:
139:
2005:
458:
1549:
1097:
941:
1971:
1909:
1623:
1284:
843:
597:
392:
302:
276:
272:
65:
29:
17:
1123:
531:
491:
1996:
1863:
1414:
25:
1580:
1976:
1562:
1457:
1447:
1362:
1981:
1899:
1868:
1848:
1833:
1828:
1823:
1660:
1554:
1515:
1484:
1404:
1396:
1044:
614:
1426:
1843:
1797:
1745:
1740:
1711:
1592:
1422:
1337:
1239:
857:
268:
261:
37:
1670:
1209:{\displaystyle \{\omega \in \Omega :F(\omega )\cap U\neq \emptyset \}\in {\mathcal {F}}}
2032:
1884:
1685:
1382:
1221:
814:
702:
682:
662:
642:
551:
511:
333:
2080:
2037:
1961:
1690:
1675:
1665:
1558:
966:
384:
2027:
1680:
1650:
935:
323:
32:. There are various selection theorems, and they are important in the theories of
279:. This is where the selection theorems come into action: they guarantee that, if
1956:
1946:
1853:
1655:
854:
587:
313:
1889:
1729:
1725:
1721:
1409:
890:
811:
In a later note, Xu proved that the
DeutschâKenderov theorem is also valid if
804:
388:
360:
356:
1566:
1461:
963:
126:
1520:
1503:
913:
842:
says that the following conditions are sufficient for the existence of a
301:
says that the following conditions are sufficient for the existence of a
1418:
1441:
1488:
1400:
1359:
Fixed Point
Theorems with Applications to Economics and Game Theory
1596:
24:
is a theorem that guarantees the existence of a single-valued
1265:
1255:
1201:
1103:
1025:
947:
407:
97:
1291:
Other selection theorems for set-valued functions include:
927:
Kuratowski and Ryll-Nardzewski measurable selection theorem
1296:
BressanâColombo directionally continuous selection theorem
776:{\textstyle \bigcap _{u\in U}\{F(u)+V\}\neq \emptyset }
1504:"A Note on a Continuous Approximate Selection Theorem"
726:
287:
that is continuous or has other desirable properties.
283:
satisfies certain properties, then it has a selection
1446:. Springer International Publishing. pp. 68â70.
1249:
1224:
1153:
1126:
1100:
1053:
1013:
975:
944:
817:
705:
685:
665:
645:
617:
554:
534:
514:
494:
461:
404:
185:
142:
82:
568:. The theorem implies the existence of a continuous
260:
returns a single value. This is a special case of a
238:{\displaystyle \forall x\in X:\,\,\,f(x)\in F(x)\,.}
2015:
1939:
1918:
1877:
1816:
1758:
1704:
1639:
1952:Spectral theory of ordinary differential equations
1275:
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1208:
1138:
1110:
1085:
1035:
995:
954:
823:
775:
711:
691:
671:
651:
629:
560:
540:
520:
500:
480:
423:
237:
160:
113:
114:{\displaystyle F:X\rightarrow {\mathcal {P}}(Y)}
1276:{\displaystyle ({\mathcal {F}},{\mathcal {B}})}
1120:measurable map (that is, for every open subset
1608:
1086:{\displaystyle F:\Omega \to \mathrm {Cl} (X)}
8:
1193:
1154:
764:
743:
1327:Selection theorems for set-valued sequences
291:Selection theorems for set-valued functions
1643:
1615:
1601:
1593:
1321:Robert Aumann measurable selection theorem
1316:Zero-dimensional Michael selection theorem
256:returns multiple values, the new function
1548:
1519:
1408:
1264:
1263:
1254:
1253:
1248:
1223:
1200:
1199:
1152:
1125:
1102:
1101:
1099:
1066:
1052:
1024:
1023:
1012:
1003:is the set of nonempty closed subsets of
976:
974:
946:
945:
943:
816:
731:
725:
704:
684:
664:
644:
616:
553:
533:
513:
493:
472:
460:
406:
405:
403:
231:
203:
202:
201:
184:
141:
96:
95:
81:
1905:Group algebra of a locally compact group
1036:{\displaystyle (\Omega ,{\mathcal {F}})}
1349:
1306:Fryszkowski decomposable map selection
1477:SIAM Journal on Mathematical Analysis
7:
1385:(1956). "Continuous selections. I".
840:Yannelis-Prabhakar selection theorem
1190:
1163:
1070:
1067:
1060:
1017:
980:
977:
770:
186:
14:
1537:Journal of Mathematical Economics
548:open balls centered on points in
424:{\displaystyle {\mathcal {P}}(Y)}
2061:
2060:
1987:Topological quantum field theory
996:{\displaystyle \mathrm {Cl} (X)}
252:for which the original function
161:{\displaystyle f:X\rightarrow Y}
2087:Theorems in functional analysis
1508:Journal of Approximation Theory
1301:Castaing representation theorem
528:, that is, the union of radius-
481:{\displaystyle _{\varepsilon }}
248:In other words, given an input
1361:. Cambridge University Press.
1270:
1250:
1178:
1172:
1111:{\displaystyle {\mathcal {F}}}
1080:
1074:
1063:
1030:
1014:
990:
984:
955:{\displaystyle {\mathcal {B}}}
755:
749:
469:
462:
418:
412:
228:
222:
213:
207:
152:
108:
102:
92:
1:
1783:Uniform boundedness principle
1502:Xu, Yuguang (December 2001).
369:approximate selection theorem
20:, a branch of mathematics, a
1559:10.1016/0304-4068(83)90041-1
1139:{\displaystyle U\subseteq X}
679:there exists a neighborhood
541:{\displaystyle \varepsilon }
501:{\displaystyle \varepsilon }
608:almost lower hemicontinuous
379:is a compact metric space,
2103:
1926:Invariant subspace problem
1333:Blaschke selection theorem
2056:
1646:
1440:Shapiro, Joel H. (2016).
1311:Helly's selection theorem
298:Michael selection theorem
1895:Spectrum of a C*-algebra
868:linear topological space
833:topological vector space
577:DeutschâKenderov theorem
1992:Noncommutative geometry
1357:Border, Kim C. (1989).
34:differential inclusions
2048:TomitaâTakesaki theory
2023:Approximation property
1967:Calculus of variations
1521:10.1006/jath.2001.3622
1277:
1232:
1210:
1140:
1112:
1087:
1037:
997:
956:
825:
777:
713:
693:
673:
653:
639:for each neighborhood
631:
630:{\displaystyle x\in X}
562:
542:
522:
502:
482:
453:
425:
239:
162:
115:
42:mathematical economics
2043:BanachâMazur distance
2006:Generalized functions
1443:A Fixed-Point Farrago
1388:Annals of Mathematics
1278:
1233:
1211:
1141:
1113:
1088:
1038:
998:
957:
826:
778:
714:
694:
674:
654:
632:
563:
543:
523:
503:
483:
426:
373:
371:states the following:
240:
163:
116:
1788:Kakutani fixed-point
1773:Riesz representation
1247:
1222:
1151:
1124:
1098:
1051:
1011:
973:
942:
831:is a locally convex
815:
724:
703:
683:
663:
643:
615:
552:
532:
512:
492:
459:
402:
334:lower hemicontinuous
183:
140:
80:
1972:Functional calculus
1931:Mahler's conjecture
1910:Von Neumann algebra
1624:Functional analysis
1583:Volume II, page 36.
610:, that is, at each
598:normed vector space
393:normed vector space
121:is a function from
66:set-valued function
18:functional analysis
1997:Riemann hypothesis
1696:Topological vector
1410:10338.dmlcz/119700
1273:
1228:
1206:
1136:
1108:
1083:
1033:
993:
952:
904:, the inverse set
889:) is nonempty and
821:
803:) is nonempty and
773:
742:
709:
689:
669:
649:
627:
558:
538:
518:
498:
478:
421:
235:
158:
111:
26:selection function
2074:
2073:
1977:Integral operator
1754:
1753:
1453:978-3-319-27978-7
1391:. Second Series.
1231:{\displaystyle F}
824:{\displaystyle Y}
727:
712:{\displaystyle x}
692:{\displaystyle U}
672:{\displaystyle 0}
652:{\displaystyle V}
561:{\displaystyle S}
521:{\displaystyle S}
22:selection theorem
2094:
2064:
2063:
1982:Jones polynomial
1900:Operator algebra
1644:
1617:
1610:
1603:
1594:
1584:
1581:"Measure Theory"
1579:V. I. Bogachev,
1577:
1571:
1570:
1552:
1532:
1526:
1525:
1523:
1499:
1493:
1492:
1472:
1466:
1465:
1437:
1431:
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1412:
1379:
1373:
1372:
1354:
1288:
1282:
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1279:
1274:
1269:
1268:
1259:
1258:
1237:
1235:
1234:
1229:
1217:
1215:
1213:
1212:
1207:
1205:
1204:
1145:
1143:
1142:
1137:
1119:
1117:
1115:
1114:
1109:
1107:
1106:
1092:
1090:
1089:
1084:
1073:
1045:measurable space
1042:
1040:
1039:
1034:
1029:
1028:
1002:
1000:
999:
994:
983:
961:
959:
958:
953:
951:
950:
830:
828:
827:
822:
784:
782:
780:
779:
774:
741:
718:
716:
715:
710:
698:
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695:
690:
678:
676:
675:
670:
658:
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638:
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634:
633:
628:
567:
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564:
559:
547:
545:
544:
539:
527:
525:
524:
519:
507:
505:
504:
499:
487:
485:
484:
479:
477:
476:
430:
428:
427:
422:
411:
410:
244:
242:
241:
236:
168:is said to be a
167:
165:
164:
159:
120:
118:
117:
112:
101:
100:
76:. Equivalently,
2102:
2101:
2097:
2096:
2095:
2093:
2092:
2091:
2077:
2076:
2075:
2070:
2052:
2016:Advanced topics
2011:
1935:
1914:
1873:
1839:HilbertâSchmidt
1812:
1803:GelfandâNaimark
1750:
1700:
1635:
1621:
1590:
1588:
1587:
1578:
1574:
1550:10.1.1.702.2938
1534:
1533:
1529:
1501:
1500:
1496:
1489:10.1137/0514015
1474:
1473:
1469:
1454:
1439:
1438:
1434:
1401:10.2307/1969615
1383:Michael, Ernest
1381:
1380:
1376:
1369:
1356:
1355:
1351:
1346:
1338:Maximum theorem
1329:
1245:
1244:
1243:
1220:
1219:
1149:
1148:
1147:
1122:
1121:
1096:
1095:
1094:
1049:
1048:
1009:
1008:
971:
970:
940:
939:
858:Hausdorff space
813:
812:
722:
721:
720:
701:
700:
681:
680:
661:
660:
641:
640:
613:
612:
611:
550:
549:
530:
529:
510:
509:
490:
489:
468:
457:
456:
450:
400:
399:
355:) is nonempty,
293:
269:axiom of choice
262:choice function
181:
180:
138:
137:
78:
77:
52:Given two sets
50:
38:optimal control
12:
11:
5:
2100:
2098:
2090:
2089:
2079:
2078:
2072:
2071:
2069:
2068:
2057:
2054:
2053:
2051:
2050:
2045:
2040:
2035:
2033:Choquet theory
2030:
2025:
2019:
2017:
2013:
2012:
2010:
2009:
1999:
1994:
1989:
1984:
1979:
1974:
1969:
1964:
1959:
1954:
1949:
1943:
1941:
1937:
1936:
1934:
1933:
1928:
1922:
1920:
1916:
1915:
1913:
1912:
1907:
1902:
1897:
1892:
1887:
1885:Banach algebra
1881:
1879:
1875:
1874:
1872:
1871:
1866:
1861:
1856:
1851:
1846:
1841:
1836:
1831:
1826:
1820:
1818:
1814:
1813:
1811:
1810:
1808:BanachâAlaoglu
1805:
1800:
1795:
1790:
1785:
1780:
1775:
1770:
1764:
1762:
1756:
1755:
1752:
1751:
1749:
1748:
1743:
1738:
1736:Locally convex
1733:
1719:
1714:
1708:
1706:
1702:
1701:
1699:
1698:
1693:
1688:
1683:
1678:
1673:
1668:
1663:
1658:
1653:
1647:
1641:
1637:
1636:
1622:
1620:
1619:
1612:
1605:
1597:
1586:
1585:
1572:
1543:(3): 233â245.
1527:
1514:(2): 324â325.
1494:
1483:(1): 185â194.
1467:
1452:
1432:
1395:(2): 361â382.
1374:
1367:
1348:
1347:
1345:
1342:
1341:
1340:
1335:
1328:
1325:
1324:
1323:
1318:
1313:
1308:
1303:
1298:
1272:
1267:
1262:
1257:
1252:
1227:
1203:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1135:
1132:
1129:
1105:
1082:
1079:
1076:
1072:
1069:
1065:
1062:
1059:
1056:
1032:
1027:
1022:
1019:
1016:
992:
989:
986:
982:
979:
949:
922:
921:
894:
871:
861:
820:
809:
808:
785:
772:
769:
766:
763:
760:
757:
754:
751:
748:
745:
740:
737:
734:
730:
708:
688:
668:
648:
626:
623:
620:
601:
591:
557:
537:
517:
497:
475:
471:
467:
464:
448:
420:
417:
414:
409:
365:
364:
337:
327:
317:
292:
289:
246:
245:
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206:
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157:
154:
151:
148:
145:
110:
107:
104:
99:
94:
91:
88:
85:
49:
46:
30:set-valued map
13:
10:
9:
6:
4:
3:
2:
2099:
2088:
2085:
2084:
2082:
2067:
2059:
2058:
2055:
2049:
2046:
2044:
2041:
2039:
2038:Weak topology
2036:
2034:
2031:
2029:
2026:
2024:
2021:
2020:
2018:
2014:
2007:
2003:
2000:
1998:
1995:
1993:
1990:
1988:
1985:
1983:
1980:
1978:
1975:
1973:
1970:
1968:
1965:
1963:
1962:Index theorem
1960:
1958:
1955:
1953:
1950:
1948:
1945:
1944:
1942:
1938:
1932:
1929:
1927:
1924:
1923:
1921:
1919:Open problems
1917:
1911:
1908:
1906:
1903:
1901:
1898:
1896:
1893:
1891:
1888:
1886:
1883:
1882:
1880:
1876:
1870:
1867:
1865:
1862:
1860:
1857:
1855:
1852:
1850:
1847:
1845:
1842:
1840:
1837:
1835:
1832:
1830:
1827:
1825:
1822:
1821:
1819:
1815:
1809:
1806:
1804:
1801:
1799:
1796:
1794:
1791:
1789:
1786:
1784:
1781:
1779:
1776:
1774:
1771:
1769:
1766:
1765:
1763:
1761:
1757:
1747:
1744:
1742:
1739:
1737:
1734:
1731:
1727:
1723:
1720:
1718:
1715:
1713:
1710:
1709:
1707:
1703:
1697:
1694:
1692:
1689:
1687:
1684:
1682:
1679:
1677:
1674:
1672:
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1667:
1664:
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1659:
1657:
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1649:
1648:
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1633:
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1613:
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1606:
1604:
1599:
1598:
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1582:
1576:
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1560:
1556:
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1528:
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1505:
1498:
1495:
1490:
1486:
1482:
1478:
1471:
1468:
1463:
1459:
1455:
1449:
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1444:
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1433:
1428:
1424:
1420:
1416:
1411:
1406:
1402:
1398:
1394:
1390:
1389:
1384:
1378:
1375:
1370:
1368:0-521-26564-9
1364:
1360:
1353:
1350:
1343:
1339:
1336:
1334:
1331:
1330:
1326:
1322:
1319:
1317:
1314:
1312:
1309:
1307:
1304:
1302:
1299:
1297:
1294:
1293:
1292:
1289:
1286:
1260:
1241:
1225:
1196:
1187:
1184:
1181:
1175:
1169:
1166:
1160:
1157:
1133:
1130:
1127:
1077:
1057:
1054:
1046:
1020:
1006:
987:
968:
965:
937:
933:
930:says that if
929:
928:
919:
915:
911:
907:
903:
899:
895:
892:
888:
884:
880:
876:
872:
869:
865:
862:
859:
856:
852:
849:
848:
847:
845:
841:
836:
834:
818:
806:
802:
798:
794:
790:
786:
767:
761:
758:
752:
746:
738:
735:
732:
728:
706:
686:
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2002:Distribution
1940:Applications
1793:KreinâMilman
1778:Closed graph
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1957:Heat kernel
1947:Hardy space
1854:Trace class
1768:HahnâBanach
1730:Topological
855:paracompact
846:selection:
588:paracompact
572:selection.
570:approximate
443:with graph(
314:paracompact
305:selection:
136:A function
1890:C*-algebra
1705:Properties
1344:References
1285:measurable
881:, the set
844:continuous
795:, the set
719:such that
347:, the set
303:continuous
273:continuity
1864:Unbounded
1859:Transpose
1817:Operators
1746:Separable
1741:Reflexive
1726:Algebraic
1712:Barrelled
1567:0304-4068
1545:CiteSeerX
1462:984777840
1240:selection
1197:∈
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1188:≠
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1176:ω
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127:power set
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2081:Category
2066:Category
1878:Algebras
1760:Theorems
1717:Complete
1686:Schwartz
1632:glossary
1242:that is
1146:we have
914:open set
912:) is an
896:for all
873:for all
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375:Suppose
339:for all
1869:Unitary
1849:Nuclear
1834:Compact
1829:Bounded
1824:Adjoint
1798:Minâmax
1691:Sobolev
1676:Nuclear
1666:Hilbert
1661:Fréchet
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1427:0077107
1419:1969615
1118:-weakly
385:compact
125:to the
1844:Normal
1681:Orlicz
1671:Hölder
1651:Banach
1640:Spaces
1628:topics
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1238:has a
1093:is an
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805:convex
590:space;
455:Here,
389:convex
361:closed
357:convex
316:space;
60:, let
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1656:Besov
1415:JSTOR
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