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183:
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77:
840:
815:
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1265:
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834:
443:. There exist σ-barrelled spaces (which are consequently σ-quasi-barrelled spaces) that are not countably quasi-barrelled spaces. There exist
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556:
1223:
1275:
772:
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450:
that are not σ-quasi-barrelled. There exist sequentially barrelled spaces that are not σ-quasi-barrelled. There exist
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1154:
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33:
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17:
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124:
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905:
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1310:
1158:
548:
463:
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220:
213:
1341:
1290:
1005:
451:
216:
155:
29:
1410:
1325:
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1178:
1138:
1066:
1041:
985:
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223:
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612:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
447:
440:
234:
1330:
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406:
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1020:
880:
762:
231:
657:
627:
593:
578:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
1295:
1112:
684:
566:
1260:
1255:
1213:
1193:
1163:
954:
473:
421:
405:
is countably quasi-barrelled and thus also σ-quasi-barrelled space. The
1203:
478:
413:
and of a metrizable locally convex space is countably quasi-barrelled.
385:
Every countably quasi-barrelled space is a σ-quasi-barrelled space.
424:
is countably quasi-barrelled. A σ-quasi-barrelled space that is
695:
523:
521:
519:
36:
is again equicontinuous. This property is a generalization of
517:
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511:
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505:
503:
501:
499:
362:
327:
292:
257:
197:
170:
139:
108:
95:
64:
454:
locally convex TVSs that are not sequentially barrelled.
230:
that is equal to the countable intersection of closed
356:
321:
286:
251:
219:
TVS is countably quasi-barrelled if and only if each
191:
164:
133:
89:
58:
667:
Schwartz spaces, nuclear spaces, and tensor products
640:
Topological Vector Spaces, Distributions and
Kernels
1131:
1075:
973:
861:
796:
730:
369:
334:
299:
264:
237:neighborhoods of 0 is itself a neighborhood of 0.
204:
177:
146:
115:
71:
116:{\displaystyle B^{\prime }\subseteq X^{\prime }}
547:. Vol. 936. Berlin, Heidelberg, New York:
574:Narici, Lawrence; Beckenstein, Edward (2011).
707:
28:if every strongly bounded countable union of
8:
541:Counterexamples in Topological Vector Spaces
527:
714:
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692:
361:
355:
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196:
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169:
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138:
132:
107:
94:
88:
63:
57:
495:
853:Uniform boundedness (Banach–Steinhaus)
154:that is equal to a countable union of
642:. Mineola, N.Y.: Dover Publications.
420:is a σ-quasi-barrelled space. Every
7:
669:. Berlin New York: Springer-Verlag.
389:Examples and sufficient conditions
311:Sequentially quasi-barrelled space
14:
315:A TVS with continuous dual space
245:A TVS with continuous dual space
1391:
1390:
1378:With the approximation property
841:Open mapping (Banach–Schauder)
1:
212:is itself equicontinuous. A
604:; Wolff, Manfred P. (1999).
545:Lecture Notes in Mathematics
344:sequentially quasi-barrelled
1062:Radially convex/Star-shaped
1047:Pre-compact/Totally bounded
539:Khaleelulla, S. M. (1982).
370:{\displaystyle X^{\prime }}
335:{\displaystyle X^{\prime }}
300:{\displaystyle X^{\prime }}
265:{\displaystyle X^{\prime }}
205:{\displaystyle B^{\prime }}
178:{\displaystyle X^{\prime }}
147:{\displaystyle X^{\prime }}
72:{\displaystyle X^{\prime }}
52:with continuous dual space
1433:
748:Continuous linear operator
1386:
1093:Algebraic interior (core)
835:Vector-valued Hahn–Banach
723:Topological vector spaces
606:Topological Vector Spaces
576:Topological Vector Spaces
469:Countably barrelled space
399:countably barrelled space
81:countably quasi-barrelled
26:countably quasi-barrelled
923:Topological homomorphism
783:Topological vector space
280:(countable) sequence in
22:topological vector space
350:convergent sequence in
241:σ-quasi-barrelled space
981:Absolutely convex/disk
371:
336:
301:
266:
206:
179:
148:
117:
73:
1016:Complemented subspace
830:hyperplane separation
445:sequentially complete
426:sequentially complete
403:quasi-barrelled space
372:
337:
302:
267:
207:
180:
149:
118:
74:
38:quasibarrelled spaces
34:continuous dual space
1266:Locally convex space
816:Closed graph theorem
768:Locally convex space
484:Quasibarrelled space
354:
319:
307:is equicontinuous.
284:
249:
189:
162:
131:
87:
56:
24:(TVS) is said to be
1417:Functional analysis
1246:Interpolation space
778:Operator topologies
602:Schaefer, Helmut H.
411:distinguished space
377:is equicontinuous.
18:functional analysis
1276:(Pseudo)Metrizable
1108:Minkowski addition
960:Sublinear function
437:σ-barrelled spaces
367:
332:
297:
262:
202:
175:
144:
113:
69:
1404:
1403:
1123:Relative interior
869:Bilinear operator
753:Linear functional
649:978-0-486-45352-1
619:978-1-4612-7155-0
558:978-3-540-11565-6
530:, pp. 28–63.
430:σ-barrelled space
418:σ-barrelled space
274:σ-quasi-barrelled
1424:
1394:
1393:
1368:Uniformly smooth
1037:
1029:
996:Balanced/Circled
986:Absorbing/Radial
716:
709:
702:
693:
688:
661:
636:Trèves, François
631:
597:
570:
531:
528:Khaleelulla 1982
525:
376:
374:
373:
368:
366:
365:
341:
339:
338:
333:
331:
330:
306:
304:
303:
298:
296:
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278:strongly bounded
271:
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211:
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201:
200:
184:
182:
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125:strongly bounded
122:
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111:
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68:
67:
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1407:
1406:
1405:
1400:
1382:
1144:B-complete/Ptak
1127:
1071:
1035:
1027:
1006:Bounding points
969:
911:Densely defined
857:
846:Bounded inverse
792:
726:
720:
677:
664:
650:
634:
620:
600:
586:
573:
559:
549:Springer-Verlag
538:
535:
534:
526:
497:
492:
464:Barrelled space
460:
395:barrelled space
391:
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357:
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351:
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165:
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84:
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53:
46:
32:subsets of its
12:
11:
5:
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1402:
1401:
1399:
1398:
1387:
1384:
1383:
1381:
1380:
1375:
1370:
1365:
1363:Ultrabarrelled
1355:
1349:
1344:
1338:
1333:
1328:
1323:
1318:
1313:
1304:
1298:
1293:
1291:Quasi-complete
1288:
1286:Quasibarrelled
1283:
1278:
1273:
1268:
1263:
1258:
1253:
1248:
1243:
1238:
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1228:
1227:
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1201:
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1129:
1128:
1126:
1125:
1115:
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1105:
1100:
1095:
1085:
1079:
1077:
1076:Set operations
1073:
1072:
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952:
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863:
859:
858:
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855:
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823:
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811:Banach–Alaoglu
808:
806:Anderson–Kadec
802:
800:
794:
793:
791:
790:
785:
780:
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734:
732:
731:Basic concepts
728:
727:
721:
719:
718:
711:
704:
696:
690:
689:
675:
662:
648:
632:
618:
598:
585:978-1584888666
584:
571:
557:
533:
532:
494:
493:
491:
488:
487:
486:
481:
476:
471:
466:
459:
456:
452:quasi-complete
390:
387:
382:
379:
364:
360:
342:is said to be
329:
325:
312:
309:
294:
290:
272:is said to be
259:
255:
242:
239:
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:
1429:
1418:
1415:
1414:
1412:
1397:
1389:
1388:
1385:
1379:
1376:
1374:
1371:
1369:
1366:
1364:
1360:
1356:
1354:) convex
1353:
1350:
1348:
1345:
1343:
1339:
1337:
1334:
1332:
1329:
1327:
1326:Semi-complete
1324:
1322:
1319:
1317:
1314:
1312:
1308:
1305:
1303:
1299:
1297:
1294:
1292:
1289:
1287:
1284:
1282:
1279:
1277:
1274:
1272:
1269:
1267:
1264:
1262:
1259:
1257:
1254:
1252:
1249:
1247:
1244:
1242:
1241:Infrabarreled
1239:
1237:
1234:
1232:
1229:
1225:
1222:
1221:
1220:
1217:
1215:
1212:
1210:
1207:
1205:
1202:
1200:
1199:Distinguished
1197:
1195:
1192:
1190:
1187:
1185:
1182:
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1177:
1175:
1171:
1167:
1165:
1162:
1160:
1156:
1152:
1150:
1147:
1145:
1142:
1140:
1137:
1136:
1134:
1132:Types of TVSs
1130:
1124:
1120:
1116:
1114:
1111:
1109:
1106:
1104:
1101:
1099:
1096:
1094:
1090:
1086:
1084:
1081:
1080:
1078:
1074:
1068:
1065:
1063:
1060:
1058:
1055:
1053:
1052:Prevalent/Shy
1050:
1048:
1045:
1043:
1042:Extreme point
1040:
1038:
1032:
1030:
1024:
1022:
1019:
1017:
1014:
1012:
1009:
1007:
1004:
1002:
999:
997:
994:
992:
989:
987:
984:
982:
979:
978:
976:
974:Types of sets
972:
966:
963:
961:
958:
956:
953:
951:
948:
944:
941:
939:
936:
934:
931:
930:
929:
926:
924:
921:
917:
916:Discontinuous
914:
912:
909:
907:
904:
902:
899:
897:
894:
892:
889:
887:
884:
883:
882:
879:
875:
872:
871:
870:
867:
866:
864:
860:
854:
851:
847:
844:
843:
842:
839:
836:
833:
831:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
803:
801:
799:
795:
789:
786:
784:
781:
779:
776:
774:
773:Metrizability
771:
769:
766:
764:
761:
759:
758:Fréchet space
756:
754:
751:
749:
746:
744:
741:
739:
736:
735:
733:
729:
724:
717:
712:
710:
705:
703:
698:
697:
694:
686:
682:
678:
676:3-540-09513-6
672:
668:
665:Wong (1979).
663:
659:
655:
651:
645:
641:
637:
633:
629:
625:
621:
615:
611:
607:
603:
599:
595:
591:
587:
581:
577:
572:
568:
564:
560:
554:
550:
546:
542:
537:
536:
529:
524:
522:
520:
518:
516:
514:
512:
510:
508:
506:
504:
502:
500:
496:
489:
485:
482:
480:
477:
475:
472:
470:
467:
465:
462:
461:
457:
455:
453:
449:
448:Mackey spaces
446:
442:
441:Mackey spaces
439:that are not
438:
433:
431:
427:
423:
419:
414:
412:
408:
404:
400:
396:
388:
386:
380:
378:
358:
349:
345:
323:
310:
308:
288:
279:
275:
253:
240:
238:
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218:
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193:
166:
157:
135:
126:
104:
100:
91:
82:
60:
51:
43:
41:
39:
35:
31:
27:
23:
19:
1302:Polynomially
1231:Grothendieck
1224:tame Fréchet
1174:Bornological
1034:Linear cone
1026:Convex cone
1001:Banach disks
943:Sesquilinear
798:Main results
788:Vector space
743:Completeness
738:Banach space
666:
639:
605:
575:
540:
435:There exist
434:
415:
401:, and every
392:
384:
343:
314:
273:
244:
227:
80:
49:
47:
25:
15:
1296:Quasinormed
1209:FK-AK space
1103:Linear span
1098:Convex hull
1083:Affine hull
886:Almost open
826:Hahn–Banach
407:strong dual
221:bornivorous
158:subsets of
1336:Stereotype
1194:(DF)-space
1189:Convenient
928:Functional
896:Continuous
881:Linear map
821:F. Riesz's
763:Linear map
490:References
381:Properties
127:subset of
44:Definition
1352:Uniformly
1311:Reflexive
1159:Barrelled
1155:Countably
1067:Symmetric
965:Transpose
658:853623322
638:(2006) .
628:840278135
594:144216834
363:′
346:if every
328:′
293:′
276:if every
258:′
214:Hausdorff
198:′
171:′
140:′
109:′
101:⊆
96:′
65:′
1411:Category
1396:Category
1347:Strictly
1321:Schwartz
1261:LF-space
1256:LB-space
1214:FK-space
1184:Complete
1164:BK-space
1089:Relative
1036:(subset)
1028:(subset)
955:Seminorm
938:Bilinear
474:DF-space
458:See also
422:DF-space
397:, every
348:strongly
235:balanced
1361:)
1309:)
1251:K-space
1236:Hilbert
1219:Fréchet
1204:F-space
1179:Brauner
1172:)
1157:)
1139:Asplund
1121:)
1091:)
1011:Bounded
906:Compact
891:Bounded
828: (
685:5126158
567:8588370
479:H-space
185:, then
1373:Webbed
1359:Quasi-
1281:Montel
1271:Mackey
1170:Ultra-
1149:Banach
1057:Radial
1021:Convex
991:Affine
933:Linear
901:Closed
725:(TVSs)
683:
673:
656:
646:
626:
616:
592:
582:
565:
555:
416:Every
393:Every
232:convex
224:barrel
48:A TVS
1331:Smith
1316:Riesz
1307:Semi-
1119:Quasi
1113:Polar
428:is a
409:of a
123:is a
950:Norm
874:form
862:Maps
681:OCLC
671:ISBN
654:OCLC
644:ISBN
624:OCLC
614:ISBN
590:OCLC
580:ISBN
563:OCLC
553:ISBN
20:, a
610:GTM
432:.
226:in
83:if
40:.
16:In
1413::
679:.
652:.
622:.
608:.
588:.
561:.
551:.
543:.
498:^
1357:(
1342:B
1340:(
1300:(
1168:(
1153:(
1117:(
1087:(
837:)
715:e
708:t
701:v
687:.
660:.
630:.
596:.
569:.
359:X
324:X
289:X
254:X
228:X
194:B
167:X
136:X
105:X
92:B
61:X
50:X
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