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77:
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1037:
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583:
409:
Every countably barrelled space is a Ï-barrelled space and every Ï-barrelled space is sequentially barrelled. Every Ï-barrelled space is a
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and thus not countably barrelled. There exist sequentially barrelled spaces that are not Ï-quasi-barrelled. There exist
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1385:
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1060:
1042:
1007:
464:
417:
33:
847:
662:
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1333:
1312:
498:
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387:
1272:
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1225:
804:
445:
17:
1257:
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628:
350:
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130:
55:
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1337:
1185:
575:
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433:
421:
344:
274:
213:
124:
37:
1368:
1317:
1032:
476:
216:
155:
29:
1437:
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1012:
964:
900:
220:
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1347:
1307:
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1017:
814:
764:
718:
639:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
468:
457:
231:
1357:
1342:
1235:
1129:
1124:
1109:
1088:
1052:
959:
779:
441:
467:
that is not a đ-barrelled space. There exist Ï-barrelled spaces that are not
1170:
1083:
1047:
907:
789:
456:
There exist Ï-barrelled spaces that are not countably barrelled. There exist
228:
684:
654:
620:
605:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
1322:
1139:
711:
593:
1287:
1282:
1240:
1220:
1190:
981:
460:
1230:
493:
399:
448:
and of a metrizable locally convex space is countably barrelled.
722:
550:
548:
546:
544:
542:
540:
538:
536:
534:
36:
is again equicontinuous. This property is a generalization of
532:
530:
528:
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524:
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520:
518:
516:
514:
359:
324:
289:
254:
197:
170:
139:
108:
95:
64:
479:
locally convex TVSs that are not sequentially barrelled.
440:
countably barrelled spaces that are not barrelled. The
227:
that is equal to the countable intersection of closed
353:
318:
283:
248:
191:
164:
133:
89:
58:
694:
Schwartz spaces, nuclear spaces, and tensor products
667:
Topological Vector Spaces, Distributions and
Kernels
1158:
1102:
1000:
888:
823:
757:
366:
331:
296:
261:
234:neighborhoods of 0 is itself a neighborhood of 0.
204:
177:
146:
115:
71:
463:that are not countably barrelled. There exists a
116:{\displaystyle B^{\prime }\subseteq X^{\prime }}
574:. Vol. 936. Berlin, Heidelberg, New York:
471:. There exist Ï-barrelled spaces that are not
219:TVS is countably barrelled if and only if each
601:Narici, Lawrence; Beckenstein, Edward (2011).
436:is countably barrelled. However, there exist
734:
8:
568:Counterexamples in Topological Vector Spaces
554:
28:if every weakly bounded countable union of
741:
727:
719:
358:
352:
323:
317:
288:
282:
253:
247:
196:
190:
169:
163:
138:
132:
107:
94:
88:
63:
57:
510:
420:that is also a đ-barrelled space is a
880:Uniform boundedness (BanachâSteinhaus)
154:that is equal to a countable union of
669:. Mineola, N.Y.: Dover Publications.
382:Every countably barrelled space is a
7:
696:. Berlin New York: Springer-Verlag.
428:Examples and sufficient conditions
14:
312:A TVS with continuous dual space
242:A TVS with continuous dual space
1418:
1417:
473:countably quasi-barrelled spaces
277:bounded (countable) sequence in
1405:With the approximation property
868:Open mapping (BanachâSchauder)
384:countably quasibarrelled space
1:
631:; Wolff, Manfred P. (1999).
572:Lecture Notes in Mathematics
396:sequentially barrelled space
308:Sequentially barrelled space
212:is itself equicontinuous. A
1089:Radially convex/Star-shaped
1074:Pre-compact/Totally bounded
566:Khaleelulla, S. M. (1982).
367:{\displaystyle X^{\prime }}
332:{\displaystyle X^{\prime }}
297:{\displaystyle X^{\prime }}
262:{\displaystyle X^{\prime }}
205:{\displaystyle B^{\prime }}
178:{\displaystyle X^{\prime }}
147:{\displaystyle X^{\prime }}
72:{\displaystyle X^{\prime }}
52:with continuous dual space
1460:
775:Continuous linear operator
1413:
1120:Algebraic interior (core)
862:Vector-valued HahnâBanach
750:Topological vector spaces
633:Topological Vector Spaces
603:Topological Vector Spaces
406:is countably barrelled.
950:Topological homomorphism
810:Topological vector space
22:topological vector space
411:Ï-quasi-barrelled space
392:Ï-quasi-barrelled space
347:convergent sequence in
1008:Absolutely convex/disk
368:
341:sequentially barrelled
333:
298:
263:
206:
179:
148:
117:
73:
1043:Complemented subspace
857:hyperplane separation
465:quasi-barrelled space
418:quasi-barrelled space
369:
334:
299:
264:
207:
180:
149:
118:
74:
34:continuous dual space
1293:Locally convex space
843:Closed graph theorem
795:Locally convex space
499:Quasibarrelled space
351:
316:
304:is equicontinuous.
281:
246:
189:
162:
131:
87:
56:
24:(TVS) is said to be
1444:Functional analysis
1273:Interpolation space
805:Operator topologies
629:Schaefer, Helmut H.
446:distinguished space
374:is equicontinuous.
81:countably barrelled
26:countably barrelled
18:functional analysis
1303:(Pseudo)Metrizable
1135:Minkowski addition
987:Sublinear function
364:
329:
294:
259:
202:
175:
144:
127:bounded subset of
113:
69:
1431:
1430:
1150:Relative interior
896:Bilinear operator
780:Linear functional
676:978-0-486-45352-1
646:978-1-4612-7155-0
585:978-3-540-11565-6
557:, pp. 28â63.
416:A locally convex
404:strong dual space
388:Ï-barrelled space
238:Ï-barrelled space
1451:
1421:
1420:
1395:Uniformly smooth
1064:
1056:
1023:Balanced/Circled
1013:Absorbing/Radial
743:
736:
729:
720:
715:
688:
663:TrÚves, François
658:
624:
597:
558:
555:Khaleelulla 1982
552:
452:Counter-examples
373:
371:
370:
365:
363:
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122:
120:
119:
114:
112:
111:
99:
98:
78:
76:
75:
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68:
67:
38:barrelled spaces
1459:
1458:
1454:
1453:
1452:
1450:
1449:
1448:
1434:
1433:
1432:
1427:
1409:
1171:B-complete/Ptak
1154:
1098:
1062:
1054:
1033:Bounding points
996:
938:Densely defined
884:
873:Bounded inverse
819:
753:
747:
704:
691:
677:
661:
647:
627:
613:
600:
586:
576:Springer-Verlag
565:
562:
561:
553:
512:
507:
489:Barrelled space
485:
454:
434:barrelled space
430:
422:barrelled space
402:is a TVS whose
380:
354:
349:
348:
319:
314:
313:
310:
284:
279:
278:
249:
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240:
192:
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165:
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159:
134:
129:
128:
103:
90:
85:
84:
59:
54:
53:
46:
32:subsets of its
12:
11:
5:
1457:
1455:
1447:
1446:
1436:
1435:
1429:
1428:
1426:
1425:
1414:
1411:
1410:
1408:
1407:
1402:
1397:
1392:
1390:Ultrabarrelled
1382:
1376:
1371:
1365:
1360:
1355:
1350:
1345:
1340:
1331:
1325:
1320:
1318:Quasi-complete
1315:
1313:Quasibarrelled
1310:
1305:
1300:
1295:
1290:
1285:
1280:
1275:
1270:
1265:
1260:
1255:
1254:
1253:
1243:
1238:
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1228:
1223:
1218:
1213:
1208:
1203:
1193:
1188:
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1173:
1168:
1162:
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1156:
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1153:
1152:
1142:
1137:
1132:
1127:
1122:
1112:
1106:
1104:
1103:Set operations
1100:
1099:
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1086:
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989:
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979:
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904:
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892:
890:
886:
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883:
882:
877:
876:
875:
865:
859:
850:
845:
840:
838:BanachâAlaoglu
835:
833:AndersonâKadec
829:
827:
821:
820:
818:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
761:
759:
758:Basic concepts
755:
754:
748:
746:
745:
738:
731:
723:
717:
716:
702:
689:
675:
659:
645:
625:
612:978-1584888666
611:
598:
584:
560:
559:
509:
508:
506:
503:
502:
501:
496:
491:
484:
481:
477:quasi-complete
453:
450:
438:semi-reflexive
429:
426:
379:
376:
361:
357:
339:is said to be
326:
322:
309:
306:
291:
287:
269:is said to be
256:
252:
239:
236:
217:locally convex
199:
195:
172:
168:
156:equicontinuous
141:
137:
110:
106:
102:
97:
93:
79:is said to be
66:
62:
45:
42:
30:equicontinuous
13:
10:
9:
6:
4:
3:
2:
1456:
1445:
1442:
1441:
1439:
1424:
1416:
1415:
1412:
1406:
1403:
1401:
1398:
1396:
1393:
1391:
1387:
1383:
1381:) convex
1380:
1377:
1375:
1372:
1370:
1366:
1364:
1361:
1359:
1356:
1354:
1353:Semi-complete
1351:
1349:
1346:
1344:
1341:
1339:
1335:
1332:
1330:
1326:
1324:
1321:
1319:
1316:
1314:
1311:
1309:
1306:
1304:
1301:
1299:
1296:
1294:
1291:
1289:
1286:
1284:
1281:
1279:
1276:
1274:
1271:
1269:
1268:Infrabarreled
1266:
1264:
1261:
1259:
1256:
1252:
1249:
1248:
1247:
1244:
1242:
1239:
1237:
1234:
1232:
1229:
1227:
1226:Distinguished
1224:
1222:
1219:
1217:
1214:
1212:
1209:
1207:
1204:
1202:
1198:
1194:
1192:
1189:
1187:
1183:
1179:
1177:
1174:
1172:
1169:
1167:
1164:
1163:
1161:
1159:Types of TVSs
1157:
1151:
1147:
1143:
1141:
1138:
1136:
1133:
1131:
1128:
1126:
1123:
1121:
1117:
1113:
1111:
1108:
1107:
1105:
1101:
1095:
1092:
1090:
1087:
1085:
1082:
1080:
1079:Prevalent/Shy
1077:
1075:
1072:
1070:
1069:Extreme point
1067:
1065:
1059:
1057:
1051:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1011:
1009:
1006:
1005:
1003:
1001:Types of sets
999:
993:
990:
988:
985:
983:
980:
978:
975:
971:
968:
966:
963:
961:
958:
957:
956:
953:
951:
948:
944:
943:Discontinuous
941:
939:
936:
934:
931:
929:
926:
924:
921:
919:
916:
914:
911:
910:
909:
906:
902:
899:
898:
897:
894:
893:
891:
887:
881:
878:
874:
871:
870:
869:
866:
863:
860:
858:
854:
851:
849:
846:
844:
841:
839:
836:
834:
831:
830:
828:
826:
822:
816:
813:
811:
808:
806:
803:
801:
800:Metrizability
798:
796:
793:
791:
788:
786:
785:Fréchet space
783:
781:
778:
776:
773:
771:
768:
766:
763:
762:
760:
756:
751:
744:
739:
737:
732:
730:
725:
724:
721:
713:
709:
705:
703:3-540-09513-6
699:
695:
692:Wong (1979).
690:
686:
682:
678:
672:
668:
664:
660:
656:
652:
648:
642:
638:
634:
630:
626:
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618:
614:
608:
604:
599:
595:
591:
587:
581:
577:
573:
569:
564:
563:
556:
551:
549:
547:
545:
543:
541:
539:
537:
535:
533:
531:
529:
527:
525:
523:
521:
519:
517:
515:
511:
504:
500:
497:
495:
492:
490:
487:
486:
482:
480:
478:
474:
470:
469:Mackey spaces
466:
462:
459:
451:
449:
447:
443:
439:
435:
427:
425:
423:
419:
414:
412:
407:
405:
401:
397:
393:
389:
385:
377:
375:
355:
346:
342:
320:
307:
305:
285:
276:
272:
250:
237:
235:
233:
230:
226:
222:
218:
215:
193:
166:
157:
135:
126:
104:
100:
91:
82:
60:
51:
43:
41:
39:
35:
31:
27:
23:
19:
1329:Polynomially
1258:Grothendieck
1251:tame Fréchet
1201:Bornological
1181:
1061:Linear cone
1053:Convex cone
1028:Banach disks
970:Sesquilinear
825:Main results
815:Vector space
770:Completeness
765:Banach space
693:
666:
632:
602:
567:
455:
431:
415:
408:
381:
340:
311:
270:
241:
224:
80:
49:
47:
25:
15:
1323:Quasinormed
1236:FK-AK space
1130:Linear span
1125:Convex hull
1110:Affine hull
913:Almost open
853:HahnâBanach
442:strong dual
271:Ï-barrelled
158:subsets of
1363:Stereotype
1221:(DF)-space
1216:Convenient
955:Functional
923:Continuous
908:Linear map
848:F. Riesz's
790:Linear map
505:References
378:Properties
44:Definition
1379:Uniformly
1338:Reflexive
1186:Barrelled
1182:Countably
1094:Symmetric
992:Transpose
685:853623322
665:(2006) .
655:840278135
621:144216834
461:DF-spaces
360:′
343:if every
325:′
290:′
273:if every
255:′
214:Hausdorff
198:′
171:′
140:′
109:′
101:⊆
96:′
65:′
1438:Category
1423:Category
1374:Strictly
1348:Schwartz
1288:LF-space
1283:LB-space
1241:FK-space
1211:Complete
1191:BK-space
1116:Relative
1063:(subset)
1055:(subset)
982:Seminorm
965:Bilinear
483:See also
394:, and a
232:balanced
1388:)
1336:)
1278:K-space
1263:Hilbert
1246:Fréchet
1231:F-space
1206:Brauner
1199:)
1184:)
1166:Asplund
1148:)
1118:)
1038:Bounded
933:Compact
918:Bounded
855: (
712:5126158
594:8588370
494:H-space
400:H-space
185:, then
1400:Webbed
1386:Quasi-
1308:Montel
1298:Mackey
1197:Ultra-
1176:Banach
1084:Radial
1048:Convex
1018:Affine
960:Linear
928:Closed
752:(TVSs)
710:
700:
683:
673:
653:
643:
619:
609:
592:
582:
458:normed
432:Every
345:weak-*
275:weak-*
229:convex
221:barrel
125:weak-*
48:A TVS
1358:Smith
1343:Riesz
1334:Semi-
1146:Quasi
1140:Polar
444:of a
398:. An
123:is a
977:Norm
901:form
889:Maps
708:OCLC
698:ISBN
681:OCLC
671:ISBN
651:OCLC
641:ISBN
617:OCLC
607:ISBN
590:OCLC
580:ISBN
390:, a
386:, a
20:, a
637:GTM
223:in
83:if
40:.
16:In
1440::
706:.
679:.
649:.
635:.
615:.
588:.
578:.
570:.
513:^
424:.
413:.
1384:(
1369:B
1367:(
1327:(
1195:(
1180:(
1144:(
1114:(
864:)
742:e
735:t
728:v
714:.
687:.
657:.
623:.
596:.
356:X
321:X
286:X
251:X
225:X
194:B
167:X
136:X
105:X
92:B
61:X
50:X
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