2683:
1968:
908:
The strong dual of a metrizable locally convex space is a DF-space but the convers is in general not true (the converse being the statement that every DF-space is the strong dual of some metrizable locally convex space). From this it follows:
1016:
There exist complete DF-spaces that are not TVS-isomorphic with the strong dual of a metrizable locally convex space. There exist DF-spaces having closed vector subspaces that are not DF-spaces.
471:
126:
201:
2004:
882:
285:
253:
158:
640:
1857:
403:
665:
518:
688:
591:
1520:
981:
961:
902:
834:
807:
765:
741:
611:
568:
548:
491:
425:
372:
337:
305:
221:
76:
690:
Consequently, a linear map from a DF-space into a locally convex space is continuous if its restriction to each bounded subset of the domain is continuous.
2131:
2106:
1683:
2088:
1810:
1665:
2556:
2058:
1997:
1641:
316:
2125:
2301:
1469:
1431:
1370:
1336:
1293:
2514:
2566:
2063:
2033:
772:
79:
44:
2712:
2686:
1990:
1533:
2474:
1622:
1513:
1397:
931:
810:
1892:
1537:
1034:
375:
1280:(1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces].
2541:
2143:
2120:
1688:
1423:
1744:
717:
2717:
2592:
1971:
1693:
1678:
1506:
1457:
1324:
1708:
58:). Grothendieck was led to introduce these spaces by the following property of strong duals of metrizable spaces: If
2101:
2096:
1953:
1713:
2649:
2186:
2038:
1907:
1831:
1948:
2445:
2255:
1764:
1062:
1007:
There exist complete DF-spaces that are not TVS-isomorphic to the strong dual of a metrizable locally convex TVS.
984:
1698:
2218:
2213:
2206:
2201:
2073:
2013:
1800:
1601:
40:
2116:
1673:
2479:
2460:
2136:
1897:
2668:
2658:
2642:
2342:
2291:
2191:
2176:
1928:
1872:
1836:
1277:
1253:
51:
1362:
430:
85:
2637:
2337:
2324:
2306:
2271:
2111:
427:
possesses a fundamental sequence of bounded (i.e. there exists a countable sequence of bounded subsets
163:
1361:. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 66 (Second ed.). Berlin, New York:
1354:
2653:
2597:
2576:
1911:
855:
258:
226:
131:
1449:
2536:
2531:
2489:
2068:
1877:
1815:
1529:
1461:
919:
616:
21:
2521:
2464:
2413:
2409:
2398:
2383:
2379:
2250:
2240:
1902:
1769:
1415:
381:
2707:
2233:
2159:
1882:
1475:
1465:
1437:
1427:
1403:
1393:
1376:
1366:
1342:
1332:
1307:
1289:
850:
694:
1059: – A generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
644:
2626:
2509:
2196:
2181:
2048:
1887:
1805:
1774:
1754:
1739:
1734:
1729:
1566:
768:
709:
698:
1303:
1269:
496:
2601:
2449:
1749:
1703:
1651:
1646:
1617:
1498:
1328:
1299:
1265:
613:
is a neighborhood of the origin if and only if for every convex, balanced, bounded subset
1576:
997:
An infinite product of non-trivial DF-spaces (i.e. all factors have non-0 dimension) is
670:
573:
2632:
2581:
2296:
1938:
1790:
1591:
1025:
966:
946:
887:
819:
792:
783:
750:
726:
596:
553:
533:
476:
410:
357:
322:
290:
206:
61:
37:
2701:
2616:
2526:
2469:
2429:
2357:
2332:
2276:
2228:
2164:
1943:
1867:
1596:
1581:
1571:
1056:
813:
2663:
2611:
2571:
2561:
2439:
2286:
2281:
2078:
2028:
1982:
1933:
1586:
1556:
1426:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
1171:"On topological spaces and topological groups with certain local countable networks
837:
776:
705:
2621:
2606:
2499:
2393:
2388:
2373:
2352:
2316:
2223:
2043:
1862:
1852:
1759:
1561:
17:
2434:
2347:
2311:
2171:
2053:
1795:
1635:
1631:
1627:
47:. They play a considerable part in the theory of topological tensor products.
1441:
1042: – Topological vector space with a complete translation-invariant metric
2586:
2403:
1479:
1346:
1311:
1407:
1380:
378:(i.e. every strongly bounded countable union of equicontinuous subsets of
2551:
2546:
2504:
2454:
2245:
1493:
1050:
1045:
744:
2494:
1039:
1170:
1004:
A closed vector subspace of a DF-space is not necessarily a DF-space.
1220:
1218:
1216:
1214:
1212:
1210:
1208:
1206:
1153:
1151:
1090:
1088:
1086:
1084:
1082:
937:
The locally convex sum of a sequence of DF-spaces is a DF-space.
922:
possessing a fundamental sequence of bounded sets is a DF-space.
1986:
1502:
1065: – tensor product defined on two topological vector spaces
1181:
1179:
940:
An inductive limit of a sequence of DF-spaces is a DF-space.
869:
390:
272:
240:
145:
987:, as well as its completion, of these spaces is a DF-space.
1030:
Pages displaying short descriptions of redirect targets
969:
949:
927:
Every
Hausdorff quotient of a DF-space is a DF-space.
890:
858:
822:
795:
753:
729:
673:
647:
619:
599:
576:
556:
536:
499:
479:
433:
413:
384:
360:
325:
293:
261:
229:
209:
166:
134:
88:
64:
1454:
Schwartz Spaces, Nuclear Spaces, and Tensor
Products
1067:
Pages displaying wikidata descriptions as a fallback
2422:
2366:
2264:
2152:
2087:
2021:
1921:
1845:
1824:
1783:
1722:
1664:
1610:
1545:
1282:
Memoirs of the
American Mathematical Society Series
43:having a property that is shared by locally convex
1858:Spectral theory of ordinary differential equations
975:
955:
896:
876:
828:
801:
759:
735:
682:
659:
634:
605:
585:
562:
542:
512:
485:
465:
419:
397:
366:
331:
299:
279:
247:
215:
195:
152:
120:
70:
1327:. Vol. 936. Berlin, Heidelberg, New York:
1998:
1514:
1288:. Providence: American Mathematical Society.
1224:
1185:
1157:
1142:
1130:
1118:
1106:
1094:
8:
1321:Counterexamples in Topological Vector Spaces
55:
1236:
1197:
128:is a sequence of convex 0-neighborhoods in
2005:
1991:
1983:
1549:
1521:
1507:
1499:
968:
948:
889:
868:
863:
857:
821:
794:
752:
728:
672:
646:
618:
598:
575:
555:
535:
504:
498:
478:
451:
438:
432:
412:
389:
383:
359:
324:
292:
271:
266:
260:
239:
234:
228:
208:
203:absorbs every strongly bounded set, then
187:
177:
165:
144:
139:
133:
106:
93:
87:
63:
1811:Group algebra of a locally compact group
1028: – Type of topological vector space
307:endowed with the strong dual topology).
1256:(1954). "Sur les espaces (F) et (DF)".
1078:
317:locally convex topological vector space
2144:Uniform boundedness (Banach–Steinhaus)
1392:. Berlin, New York: Springer-Verlag.
7:
45:metrizable topological vector spaces
667:is a neighborhood of the origin in
466:{\displaystyle B_{1},B_{2},\ldots }
121:{\displaystyle V_{1},V_{2},\ldots }
1460:. Vol. 726. Berlin New York:
473:such that every bounded subset of
14:
915:Every Banach space is a DF-space.
912:Every normed space is a DF-space.
196:{\displaystyle V:=\cap _{i}V_{i}}
54:and studied in detail by him in (
2682:
2681:
1967:
1966:
1893:Topological quantum field theory
1053: – Topological vector space
287:is the continuous dual space of
50:DF-spaces were first defined by
2669:With the approximation property
1035:Countably quasi-barrelled space
877:{\displaystyle X_{b}^{\prime }}
570:be a convex balanced subset of
376:countably quasi-barrelled space
280:{\displaystyle X_{b}^{\prime }}
248:{\displaystyle X_{b}^{\prime }}
153:{\displaystyle X_{b}^{\prime }}
2132:Open mapping (Banach–Schauder)
1:
1689:Uniform boundedness principle
1390:Nuclear locally convex spaces
1359:Nuclear Locally Convex Spaces
635:{\displaystyle B\subseteq X,}
1458:Lecture Notes in Mathematics
1418:; Wolff, Manfred P. (1999).
1325:Lecture Notes in Mathematics
934:of a DF-space is a DF-space.
2353:Radially convex/Star-shaped
2338:Pre-compact/Totally bounded
1319:Khaleelulla, S. M. (1982).
743:is either a DF-space or an
704:Every infinite-dimensional
398:{\displaystyle X^{\prime }}
2734:
2039:Continuous linear operator
1832:Invariant subspace problem
1388:Pietsch, Albrecht (1972).
2713:Topological vector spaces
2677:
2384:Algebraic interior (core)
2126:Vector-valued Hahn–Banach
2014:Topological vector spaces
1962:
1552:
1420:Topological Vector Spaces
1225:Schaefer & Wolff 1999
1186:Schaefer & Wolff 1999
1158:Schaefer & Wolff 1999
1143:Schaefer & Wolff 1999
1131:Schaefer & Wolff 1999
1119:Schaefer & Wolff 1999
1107:Schaefer & Wolff 1999
1095:Schaefer & Wolff 1999
1063:Projective tensor product
985:projective tensor product
82:locally convex space and
2214:Topological homomorphism
2074:Topological vector space
1801:Spectrum of a C*-algebra
983:are DF-spaces. Then the
41:topological vector space
1898:Noncommutative geometry
1278:Grothendieck, Alexander
1254:Grothendieck, Alexander
660:{\displaystyle B\cap V}
405:is equicontinuous), and
223:is a 0-neighborhood in
2272:Absolutely convex/disk
1954:Tomita–Takesaki theory
1929:Approximation property
1873:Calculus of variations
977:
957:
898:
878:
830:
803:
761:
737:
684:
661:
636:
607:
587:
564:
550:be a DF-space and let
544:
514:
487:
467:
421:
399:
368:
333:
301:
281:
249:
217:
197:
154:
122:
72:
52:Alexander Grothendieck
2307:Complemented subspace
2121:hyperplane separation
1949:Banach–Mazur distance
1912:Generalized functions
978:
958:
899:
879:
845:Sufficient conditions
831:
804:
786:DF-space is complete.
762:
738:
718:Fréchet–Urysohn space
685:
662:
637:
608:
588:
565:
545:
515:
513:{\displaystyle B_{i}}
493:is contained in some
488:
468:
422:
400:
369:
334:
302:
282:
250:
218:
198:
155:
123:
73:
2557:Locally convex space
2107:Closed graph theorem
2059:Locally convex space
1694:Kakutani fixed-point
1679:Riesz representation
1109:, pp. 152, 154.
967:
947:
888:
856:
820:
793:
751:
727:
671:
645:
617:
597:
574:
554:
534:
497:
477:
431:
411:
382:
358:
323:
291:
259:
227:
207:
164:
132:
86:
62:
2718:Functional analysis
2537:Interpolation space
2069:Operator topologies
1878:Functional calculus
1837:Mahler's conjecture
1816:Von Neumann algebra
1530:Functional analysis
1494:DF-space at ncatlab
1416:Schaefer, Helmut H.
1258:Summa Brasil. Math.
1239:, pp. 103–110.
1227:, pp. 196–197.
1160:, pp. 199–202.
1145:, pp. 190–202.
1097:, pp. 154–155.
920:infrabarreled space
884:of a Fréchet space
873:
697:of a DF-space is a
276:
244:
149:
22:functional analysis
2567:(Pseudo)Metrizable
2399:Minkowski addition
2251:Sublinear function
1903:Riemann hypothesis
1602:Topological vector
1169:Gabriyelyan, S.S.
973:
953:
894:
874:
859:
826:
799:
771:then it is either
757:
733:
683:{\displaystyle B.}
680:
657:
632:
603:
586:{\displaystyle X.}
583:
560:
540:
510:
483:
463:
417:
395:
364:
329:
297:
277:
262:
245:
230:
213:
193:
150:
135:
118:
68:
2695:
2694:
2414:Relative interior
2160:Bilinear operator
2044:Linear functional
1980:
1979:
1883:Integral operator
1660:
1659:
1471:978-3-540-09513-2
1433:978-1-4612-7155-0
1372:978-0-387-05644-9
1355:Pietsch, Albrecht
1338:978-3-540-11565-6
1295:978-0-8218-1216-7
976:{\displaystyle Y}
956:{\displaystyle X}
897:{\displaystyle X}
851:strong dual space
829:{\displaystyle X}
802:{\displaystyle X}
760:{\displaystyle X}
736:{\displaystyle X}
695:strong dual space
606:{\displaystyle V}
563:{\displaystyle V}
543:{\displaystyle X}
486:{\displaystyle X}
420:{\displaystyle X}
367:{\displaystyle X}
332:{\displaystyle X}
300:{\displaystyle X}
216:{\displaystyle V}
71:{\displaystyle X}
56:Grothendieck 1954
2725:
2685:
2684:
2659:Uniformly smooth
2328:
2320:
2287:Balanced/Circled
2277:Absorbing/Radial
2007:
2000:
1993:
1984:
1970:
1969:
1888:Jones polynomial
1806:Operator algebra
1550:
1523:
1516:
1509:
1500:
1483:
1445:
1411:
1384:
1350:
1315:
1273:
1240:
1237:Khaleelulla 1982
1234:
1228:
1222:
1201:
1198:Khaleelulla 1982
1195:
1189:
1183:
1174:
1167:
1161:
1155:
1146:
1140:
1134:
1128:
1122:
1116:
1110:
1104:
1098:
1092:
1068:
1031:
982:
980:
979:
974:
962:
960:
959:
954:
903:
901:
900:
895:
883:
881:
880:
875:
872:
867:
835:
833:
832:
827:
808:
806:
805:
800:
769:sequential space
766:
764:
763:
758:
742:
740:
739:
734:
710:sequential space
689:
687:
686:
681:
666:
664:
663:
658:
641:
639:
638:
633:
612:
610:
609:
604:
592:
590:
589:
584:
569:
567:
566:
561:
549:
547:
546:
541:
519:
517:
516:
511:
509:
508:
492:
490:
489:
484:
472:
470:
469:
464:
456:
455:
443:
442:
426:
424:
423:
418:
404:
402:
401:
396:
394:
393:
373:
371:
370:
365:
338:
336:
335:
330:
306:
304:
303:
298:
286:
284:
283:
278:
275:
270:
254:
252:
251:
246:
243:
238:
222:
220:
219:
214:
202:
200:
199:
194:
192:
191:
182:
181:
159:
157:
156:
151:
148:
143:
127:
125:
124:
119:
111:
110:
98:
97:
77:
75:
74:
69:
2733:
2732:
2728:
2727:
2726:
2724:
2723:
2722:
2698:
2697:
2696:
2691:
2673:
2435:B-complete/Ptak
2418:
2362:
2326:
2318:
2297:Bounding points
2260:
2202:Densely defined
2148:
2137:Bounded inverse
2083:
2017:
2011:
1981:
1976:
1958:
1922:Advanced topics
1917:
1841:
1820:
1779:
1745:Hilbert–Schmidt
1718:
1709:Gelfand–Naimark
1656:
1606:
1541:
1527:
1490:
1472:
1462:Springer-Verlag
1450:Wong, Yau-Chuen
1448:
1434:
1414:
1400:
1387:
1373:
1363:Springer-Verlag
1353:
1339:
1329:Springer-Verlag
1318:
1296:
1276:
1252:
1249:
1244:
1243:
1235:
1231:
1223:
1204:
1196:
1192:
1184:
1177:
1168:
1164:
1156:
1149:
1141:
1137:
1129:
1125:
1117:
1113:
1105:
1101:
1093:
1080:
1075:
1066:
1029:
1022:
1014:
965:
964:
945:
944:
904:is a DF-space.
886:
885:
854:
853:
847:
818:
817:
791:
790:
749:
748:
725:
724:
669:
668:
643:
642:
615:
614:
595:
594:
572:
571:
552:
551:
532:
531:
527:
500:
495:
494:
475:
474:
447:
434:
429:
428:
409:
408:
385:
380:
379:
356:
355:
343:, also written
321:
320:
313:
289:
288:
257:
256:
225:
224:
205:
204:
183:
173:
162:
161:
130:
129:
102:
89:
84:
83:
60:
59:
28:, also written
12:
11:
5:
2731:
2729:
2721:
2720:
2715:
2710:
2700:
2699:
2693:
2692:
2690:
2689:
2678:
2675:
2674:
2672:
2671:
2666:
2661:
2656:
2654:Ultrabarrelled
2646:
2640:
2635:
2629:
2624:
2619:
2614:
2609:
2604:
2595:
2589:
2584:
2582:Quasi-complete
2579:
2577:Quasibarrelled
2574:
2569:
2564:
2559:
2554:
2549:
2544:
2539:
2534:
2529:
2524:
2519:
2518:
2517:
2507:
2502:
2497:
2492:
2487:
2482:
2477:
2472:
2467:
2457:
2452:
2442:
2437:
2432:
2426:
2424:
2420:
2419:
2417:
2416:
2406:
2401:
2396:
2391:
2386:
2376:
2370:
2368:
2367:Set operations
2364:
2363:
2361:
2360:
2355:
2350:
2345:
2340:
2335:
2330:
2322:
2314:
2309:
2304:
2299:
2294:
2289:
2284:
2279:
2274:
2268:
2266:
2262:
2261:
2259:
2258:
2253:
2248:
2243:
2238:
2237:
2236:
2231:
2226:
2216:
2211:
2210:
2209:
2204:
2199:
2194:
2189:
2184:
2179:
2169:
2168:
2167:
2156:
2154:
2150:
2149:
2147:
2146:
2141:
2140:
2139:
2129:
2123:
2114:
2109:
2104:
2102:Banach–Alaoglu
2099:
2097:Anderson–Kadec
2093:
2091:
2085:
2084:
2082:
2081:
2076:
2071:
2066:
2061:
2056:
2051:
2046:
2041:
2036:
2031:
2025:
2023:
2022:Basic concepts
2019:
2018:
2012:
2010:
2009:
2002:
1995:
1987:
1978:
1977:
1975:
1974:
1963:
1960:
1959:
1957:
1956:
1951:
1946:
1941:
1939:Choquet theory
1936:
1931:
1925:
1923:
1919:
1918:
1916:
1915:
1905:
1900:
1895:
1890:
1885:
1880:
1875:
1870:
1865:
1860:
1855:
1849:
1847:
1843:
1842:
1840:
1839:
1834:
1828:
1826:
1822:
1821:
1819:
1818:
1813:
1808:
1803:
1798:
1793:
1791:Banach algebra
1787:
1785:
1781:
1780:
1778:
1777:
1772:
1767:
1762:
1757:
1752:
1747:
1742:
1737:
1732:
1726:
1724:
1720:
1719:
1717:
1716:
1714:Banach–Alaoglu
1711:
1706:
1701:
1696:
1691:
1686:
1681:
1676:
1670:
1668:
1662:
1661:
1658:
1657:
1655:
1654:
1649:
1644:
1642:Locally convex
1639:
1625:
1620:
1614:
1612:
1608:
1607:
1605:
1604:
1599:
1594:
1589:
1584:
1579:
1574:
1569:
1564:
1559:
1553:
1547:
1543:
1542:
1528:
1526:
1525:
1518:
1511:
1503:
1497:
1496:
1489:
1488:External links
1486:
1485:
1484:
1470:
1446:
1432:
1412:
1398:
1385:
1371:
1351:
1337:
1316:
1294:
1274:
1248:
1245:
1242:
1241:
1229:
1202:
1190:
1188:, p. 154.
1175:
1162:
1147:
1135:
1133:, p. 196.
1123:
1111:
1099:
1077:
1076:
1074:
1071:
1070:
1069:
1060:
1054:
1048:
1043:
1037:
1032:
1026:Barreled space
1021:
1018:
1013:
1010:
1009:
1008:
1005:
1002:
1000:
991:
990:
988:
972:
952:
941:
938:
935:
928:
925:
924:
923:
916:
913:
893:
871:
866:
862:
846:
843:
842:
841:
825:
816:DF-space then
798:
787:
784:quasi-complete
780:
756:
732:
721:
715:
708:DF-space is a
702:
691:
679:
676:
656:
653:
650:
631:
628:
625:
622:
602:
582:
579:
559:
539:
526:
523:
522:
521:
507:
503:
482:
462:
459:
454:
450:
446:
441:
437:
416:
406:
392:
388:
363:
328:
312:
309:
296:
274:
269:
265:
242:
237:
233:
212:
190:
186:
180:
176:
172:
169:
147:
142:
138:
117:
114:
109:
105:
101:
96:
92:
67:
38:locally convex
13:
10:
9:
6:
4:
3:
2:
2730:
2719:
2716:
2714:
2711:
2709:
2706:
2705:
2703:
2688:
2680:
2679:
2676:
2670:
2667:
2665:
2662:
2660:
2657:
2655:
2651:
2647:
2645:) convex
2644:
2641:
2639:
2636:
2634:
2630:
2628:
2625:
2623:
2620:
2618:
2617:Semi-complete
2615:
2613:
2610:
2608:
2605:
2603:
2599:
2596:
2594:
2590:
2588:
2585:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2563:
2560:
2558:
2555:
2553:
2550:
2548:
2545:
2543:
2540:
2538:
2535:
2533:
2532:Infrabarreled
2530:
2528:
2525:
2523:
2520:
2516:
2513:
2512:
2511:
2508:
2506:
2503:
2501:
2498:
2496:
2493:
2491:
2490:Distinguished
2488:
2486:
2483:
2481:
2478:
2476:
2473:
2471:
2468:
2466:
2462:
2458:
2456:
2453:
2451:
2447:
2443:
2441:
2438:
2436:
2433:
2431:
2428:
2427:
2425:
2423:Types of TVSs
2421:
2415:
2411:
2407:
2405:
2402:
2400:
2397:
2395:
2392:
2390:
2387:
2385:
2381:
2377:
2375:
2372:
2371:
2369:
2365:
2359:
2356:
2354:
2351:
2349:
2346:
2344:
2343:Prevalent/Shy
2341:
2339:
2336:
2334:
2333:Extreme point
2331:
2329:
2323:
2321:
2315:
2313:
2310:
2308:
2305:
2303:
2300:
2298:
2295:
2293:
2290:
2288:
2285:
2283:
2280:
2278:
2275:
2273:
2270:
2269:
2267:
2265:Types of sets
2263:
2257:
2254:
2252:
2249:
2247:
2244:
2242:
2239:
2235:
2232:
2230:
2227:
2225:
2222:
2221:
2220:
2217:
2215:
2212:
2208:
2207:Discontinuous
2205:
2203:
2200:
2198:
2195:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2174:
2173:
2170:
2166:
2163:
2162:
2161:
2158:
2157:
2155:
2151:
2145:
2142:
2138:
2135:
2134:
2133:
2130:
2127:
2124:
2122:
2118:
2115:
2113:
2110:
2108:
2105:
2103:
2100:
2098:
2095:
2094:
2092:
2090:
2086:
2080:
2077:
2075:
2072:
2070:
2067:
2065:
2064:Metrizability
2062:
2060:
2057:
2055:
2052:
2050:
2049:Fréchet space
2047:
2045:
2042:
2040:
2037:
2035:
2032:
2030:
2027:
2026:
2024:
2020:
2015:
2008:
2003:
2001:
1996:
1994:
1989:
1988:
1985:
1973:
1965:
1964:
1961:
1955:
1952:
1950:
1947:
1945:
1944:Weak topology
1942:
1940:
1937:
1935:
1932:
1930:
1927:
1926:
1924:
1920:
1913:
1909:
1906:
1904:
1901:
1899:
1896:
1894:
1891:
1889:
1886:
1884:
1881:
1879:
1876:
1874:
1871:
1869:
1868:Index theorem
1866:
1864:
1861:
1859:
1856:
1854:
1851:
1850:
1848:
1844:
1838:
1835:
1833:
1830:
1829:
1827:
1825:Open problems
1823:
1817:
1814:
1812:
1809:
1807:
1804:
1802:
1799:
1797:
1794:
1792:
1789:
1788:
1786:
1782:
1776:
1773:
1771:
1768:
1766:
1763:
1761:
1758:
1756:
1753:
1751:
1748:
1746:
1743:
1741:
1738:
1736:
1733:
1731:
1728:
1727:
1725:
1721:
1715:
1712:
1710:
1707:
1705:
1702:
1700:
1697:
1695:
1692:
1690:
1687:
1685:
1682:
1680:
1677:
1675:
1672:
1671:
1669:
1667:
1663:
1653:
1650:
1648:
1645:
1643:
1640:
1637:
1633:
1629:
1626:
1624:
1621:
1619:
1616:
1615:
1613:
1609:
1603:
1600:
1598:
1595:
1593:
1590:
1588:
1585:
1583:
1580:
1578:
1575:
1573:
1570:
1568:
1565:
1563:
1560:
1558:
1555:
1554:
1551:
1548:
1544:
1539:
1535:
1531:
1524:
1519:
1517:
1512:
1510:
1505:
1504:
1501:
1495:
1492:
1491:
1487:
1481:
1477:
1473:
1467:
1463:
1459:
1455:
1451:
1447:
1443:
1439:
1435:
1429:
1425:
1421:
1417:
1413:
1409:
1405:
1401:
1399:0-387-05644-0
1395:
1391:
1386:
1382:
1378:
1374:
1368:
1364:
1360:
1356:
1352:
1348:
1344:
1340:
1334:
1330:
1326:
1322:
1317:
1313:
1309:
1305:
1301:
1297:
1291:
1287:
1284:(in French).
1283:
1279:
1275:
1271:
1267:
1263:
1260:(in French).
1259:
1255:
1251:
1250:
1246:
1238:
1233:
1230:
1226:
1221:
1219:
1217:
1215:
1213:
1211:
1209:
1207:
1203:
1200:, p. 33.
1199:
1194:
1191:
1187:
1182:
1180:
1176:
1172:
1166:
1163:
1159:
1154:
1152:
1148:
1144:
1139:
1136:
1132:
1127:
1124:
1121:, p. 25.
1120:
1115:
1112:
1108:
1103:
1100:
1096:
1091:
1089:
1087:
1085:
1083:
1079:
1072:
1064:
1061:
1058:
1057:Nuclear space
1055:
1052:
1049:
1047:
1044:
1041:
1038:
1036:
1033:
1027:
1024:
1023:
1019:
1017:
1011:
1006:
1003:
998:
996:
995:
994:
989:
986:
970:
950:
943:Suppose that
942:
939:
936:
933:
929:
926:
921:
917:
914:
911:
910:
907:
906:
905:
891:
864:
860:
852:
844:
839:
823:
815:
812:
796:
788:
785:
781:
778:
774:
770:
754:
746:
730:
722:
719:
713:
711:
707:
703:
700:
699:Fréchet space
696:
692:
677:
674:
654:
651:
648:
629:
626:
623:
620:
600:
580:
577:
557:
537:
529:
528:
524:
505:
501:
480:
460:
457:
452:
448:
444:
439:
435:
414:
407:
386:
377:
361:
354:
353:
352:
350:
348:
342:
326:
318:
310:
308:
294:
267:
263:
235:
231:
210:
188:
184:
178:
174:
170:
167:
140:
136:
115:
112:
107:
103:
99:
94:
90:
81:
65:
57:
53:
48:
46:
42:
39:
35:
33:
27:
23:
19:
2593:Polynomially
2522:Grothendieck
2515:tame Fréchet
2484:
2465:Bornological
2325:Linear cone
2317:Convex cone
2292:Banach disks
2234:Sesquilinear
2089:Main results
2079:Vector space
2034:Completeness
2029:Banach space
1934:Balanced set
1908:Distribution
1846:Applications
1699:Krein–Milman
1684:Closed graph
1453:
1419:
1389:
1358:
1320:
1285:
1281:
1261:
1257:
1247:Bibliography
1232:
1193:
1165:
1138:
1126:
1114:
1102:
1015:
992:
848:
838:Montel space
777:Montel space
346:
344:
340:
314:
49:
31:
29:
25:
18:mathematical
15:
2587:Quasinormed
2500:FK-AK space
2394:Linear span
2389:Convex hull
2374:Affine hull
2177:Almost open
2117:Hahn–Banach
1863:Heat kernel
1853:Hardy space
1760:Trace class
1674:Hahn–Banach
1636:Topological
1001:a DF-space.
2702:Categories
2627:Stereotype
2485:(DF)-space
2480:Convenient
2219:Functional
2187:Continuous
2172:Linear map
2112:F. Riesz's
2054:Linear map
1796:C*-algebra
1611:Properties
1264:: 57–123.
932:completion
775:or else a
773:metrizable
525:Properties
311:Definition
160:such that
80:metrizable
2643:Uniformly
2602:Reflexive
2450:Barrelled
2446:Countably
2358:Symmetric
2256:Transpose
1770:Unbounded
1765:Transpose
1723:Operators
1652:Separable
1647:Reflexive
1632:Algebraic
1618:Barrelled
1442:840278135
1073:Citations
993:However,
870:′
779:DF-space.
652:∩
624:⊆
461:…
391:′
273:′
241:′
175:∩
146:′
116:…
26:DF-spaces
20:field of
2708:Topology
2687:Category
2638:Strictly
2612:Schwartz
2552:LF-space
2547:LB-space
2505:FK-space
2475:Complete
2455:BK-space
2380:Relative
2327:(subset)
2319:(subset)
2246:Seminorm
2229:Bilinear
1972:Category
1784:Algebras
1666:Theorems
1623:Complete
1592:Schwartz
1538:glossary
1452:(1979).
1357:(1979).
1051:LF-space
1046:LB-space
1020:See also
1012:Examples
811:complete
745:LM-space
723:Suppose
341:DF-space
34:)-spaces
2652:)
2600:)
2542:K-space
2527:Hilbert
2510:Fréchet
2495:F-space
2470:Brauner
2463:)
2448:)
2430:Asplund
2412:)
2382:)
2302:Bounded
2197:Compact
2182:Bounded
2119: (
1775:Unitary
1755:Nuclear
1740:Compact
1735:Bounded
1730:Adjoint
1704:Min–max
1597:Sobolev
1582:Nuclear
1572:Hilbert
1567:Fréchet
1532: (
1480:5126158
1347:8588370
1312:1315788
1304:0075539
1270:0075542
1040:F-space
814:nuclear
349:)-space
255:(where
16:In the
2664:Webbed
2650:Quasi-
2572:Montel
2562:Mackey
2461:Ultra-
2440:Banach
2348:Radial
2312:Convex
2282:Affine
2224:Linear
2192:Closed
2016:(TVSs)
1750:Normal
1587:Orlicz
1577:Hölder
1557:Banach
1546:Spaces
1534:topics
1478:
1468:
1440:
1430:
1408:539541
1406:
1396:
1381:539541
1379:
1369:
1345:
1335:
1310:
1302:
1292:
1268:
1173:(2014)
918:Every
782:Every
706:Montel
319:(TVS)
2622:Smith
2607:Riesz
2598:Semi-
2410:Quasi
2404:Polar
1562:Besov
836:is a
809:is a
767:is a
747:. If
593:Then
374:is a
351:, if
339:is a
78:is a
2241:Norm
2165:form
2153:Maps
1910:(or
1628:Dual
1476:OCLC
1466:ISBN
1438:OCLC
1428:ISBN
1404:OCLC
1394:ISBN
1377:OCLC
1367:ISBN
1343:OCLC
1333:ISBN
1308:OCLC
1290:ISBN
963:and
930:The
849:The
712:but
693:The
530:Let
36:are
1424:GTM
999:not
789:If
714:not
2704::
1536:–
1474:.
1464:.
1456:.
1436:.
1422:.
1402:.
1375:.
1365:.
1341:.
1331:.
1323:.
1306:.
1300:MR
1298:.
1286:16
1266:MR
1205:^
1178:^
1150:^
1081:^
716:a
520:).
347:DF
315:A
171::=
32:DF
24:,
2648:(
2633:B
2631:(
2591:(
2459:(
2444:(
2408:(
2378:(
2128:)
2006:e
1999:t
1992:v
1914:)
1638:)
1634:/
1630:(
1540:)
1522:e
1515:t
1508:v
1482:.
1444:.
1410:.
1383:.
1349:.
1314:.
1272:.
1262:3
971:Y
951:X
892:X
865:b
861:X
840:.
824:X
797:X
755:X
731:X
720:.
701:.
678:.
675:B
655:V
649:B
630:,
627:X
621:B
601:V
581:.
578:X
558:V
538:X
506:i
502:B
481:X
458:,
453:2
449:B
445:,
440:1
436:B
415:X
387:X
362:X
345:(
327:X
295:X
268:b
264:X
236:b
232:X
211:V
189:i
185:V
179:i
168:V
141:b
137:X
113:,
108:2
104:V
100:,
95:1
91:V
66:X
30:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.