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Every continuous linear map from a Ptak space onto a barreled space is a topological homomorphism.
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1654:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
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1620:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
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2190:
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of a Ptak space is a Ptak space. If every
Hausdorff quotient of a TVS
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is a locally convex space such that there exists a continuous
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Pages displaying short descriptions of redirect targets
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Every closed vector subspace of a Ptak space (resp. a B
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be a nearly open linear map whose domain is dense in a
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185:{\displaystyle \sigma \left(X^{\prime },X\right)}
1589:. Vol. 936. Berlin, Heidelberg, New York:
1554:. Vol. 692. Berlin, New York, Heidelberg:
832:is dense in some neighborhood of the origin in
1616:Narici, Lawrence; Beckenstein, Edward (2011).
1704:
1529:
1514:
1499:
1484:
8:
1583:Counterexamples in Topological Vector Spaces
1208:-complete spaces that are not B-complete.
1546:Husain, Taqdir; Khaleelulla, S. M. (1978).
993:
985:. However, there exist complete Hausdorff
1711:
1697:
1689:
1226:-complete space) is a Ptak space (resp. a
1070:and whose range is a locally convex space
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1454: – Type of topological vector space
1472:
1215:is a Ptak space. The strong dual of a
1850:Uniform boundedness (Banach–Steinhaus)
603:{\displaystyle A\subseteq X^{\prime }}
440:{\displaystyle Q\subseteq X^{\prime }}
316:{\displaystyle A\subseteq X^{\prime }}
80:{\displaystyle Q\subseteq X^{\prime }}
570:{\displaystyle X_{\sigma }^{\prime }}
472:{\displaystyle X_{\sigma }^{\prime }}
283:{\displaystyle X_{\sigma }^{\prime }}
139:{\displaystyle X_{\sigma }^{\prime }}
7:
545:is given the subspace topology from
258:is given the subspace topology from
87:is closed in the weak-* topology on
1200:Examples and sufficient conditions
14:
577:) for each equicontinuous subset
290:) for each equicontinuous subset
2388:
2387:
989:space that are not Ptak spaces.
2375:With the approximation property
1219:Fréchet space is a Ptak space.
871:The following are equivalent:
648:The following are equivalent:
1838:Open mapping (Banach–Schauder)
1351:
952:into any locally convex space
848:
842:
819:
813:
743:
718:is a topological homomorphism.
698:into any locally convex space
1:
1646:; Wolff, Manfred P. (1999).
1587:Lecture Notes in Mathematics
1552:Lecture Notes in Mathematics
1253:-complete space). and every
1090:. Suppose that the graph of
329:-completeness is related to
2059:Radially convex/Star-shaped
2044:Pre-compact/Totally bounded
1581:Khaleelulla, S. M. (1982).
1414:that is B-complete (resp. B
107:{\displaystyle X^{\prime }}
2430:
1745:Continuous linear operator
2414:Topological vector spaces
2383:
2090:Algebraic interior (core)
1832:Vector-valued Hahn–Banach
1720:Topological vector spaces
1648:Topological Vector Spaces
1618:Topological Vector Spaces
1530:Schaefer & Wolff 1999
1515:Schaefer & Wolff 1999
1500:Schaefer & Wolff 1999
1485:Schaefer & Wolff 1999
1129:{\displaystyle X\times Y}
763:if for each neighborhood
618:Throughout this section,
1920:Topological homomorphism
1780:Topological vector space
1684:Nuclear space at ncatlab
1367:from a Ptak space, then
1360:{\displaystyle u:P\to X}
752:{\displaystyle u:X\to Y}
643:topological vector space
21:topological vector space
498:{\displaystyle Q\cap A}
356:-completeness, where a
211:{\displaystyle Q\cap A}
1978:Absolutely convex/disk
1438:is B-complete (resp. B
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2013:Complemented subspace
1827:hyperplane separation
1433:
1405:
1382:
1362:
1326:
1299:
1284:-complete space then
1275:
1248:
1246:{\displaystyle B_{r}}
1191:
1176:is a Ptak space then
1171:
1151:
1131:
1105:
1085:
1065:
1045:
1043:{\displaystyle B_{r}}
1018:
972:is a TVS-isomorphism.
967:
947:
916:
914:{\displaystyle B_{r}}
889:
859:
857:{\displaystyle u(X).}
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401:{\displaystyle B_{r}}
375:
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349:{\displaystyle B_{r}}
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2263:Locally convex space
1813:Closed graph theorem
1765:Locally convex space
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1007:
994:Homomorphism Theorem
981:Every Ptak space is
956:
936:
898:
878:
836:
825:{\displaystyle u(U)}
807:
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27:
2243:Interpolation space
1775:Operator topologies
1644:Schaefer, Helmut H.
1387:is a Ptak space.
1308:-complete space.
1156:is injective or if
997: —
566:
468:
279:
135:
2273:(Pseudo)Metrizable
2105:Minkowski addition
1957:Sublinear function
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54:if every subspace
33:
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2120:Relative interior
1866:Bilinear operator
1750:Linear functional
1661:978-1-4612-7155-0
1600:978-3-540-11565-6
1565:978-3-540-09096-0
1431:{\displaystyle X}
1403:{\displaystyle X}
1380:{\displaystyle X}
1324:{\displaystyle X}
1297:{\displaystyle X}
1273:{\displaystyle X}
1189:{\displaystyle u}
1169:{\displaystyle X}
1149:{\displaystyle u}
1103:{\displaystyle u}
1083:{\displaystyle Y}
1063:{\displaystyle X}
1016:{\displaystyle u}
965:{\displaystyle Y}
945:{\displaystyle X}
924:Every continuous
887:{\displaystyle X}
796:{\displaystyle X}
783:of the origin in
776:{\displaystyle U}
711:{\displaystyle Y}
691:{\displaystyle X}
674:Every continuous
664:{\displaystyle X}
631:{\displaystyle X}
614:Characterizations
538:{\displaystyle A}
518:{\displaystyle A}
373:{\displaystyle X}
251:{\displaystyle A}
231:{\displaystyle A}
36:{\displaystyle X}
2421:
2391:
2390:
2365:Uniformly smooth
2034:
2026:
1993:Balanced/Circled
1983:Absorbing/Radial
1713:
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1418:-complete) then
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1050:-complete space
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2141:B-complete/Ptak
2124:
2068:
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2024:
2003:Bounding points
1966:
1908:Densely defined
1854:
1843:Bounded inverse
1789:
1723:
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1680:
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1591:Springer-Verlag
1580:
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1556:Springer-Verlag
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2360:Ultrabarrelled
2352:
2346:
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2320:
2315:
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2301:
2295:
2290:
2288:Quasi-complete
2285:
2283:Quasibarrelled
2280:
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2102:
2097:
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2076:
2074:
2073:Set operations
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2069:
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2020:
2015:
2010:
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1810:
1808:Banach–Alaoglu
1805:
1803:Anderson–Kadec
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1777:
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1729:
1728:Basic concepts
1725:
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1686:
1679:
1678:External links
1676:
1675:
1674:
1660:
1640:
1627:978-1584888666
1626:
1613:
1599:
1578:
1564:
1541:
1538:
1535:
1534:
1532:, p. 165.
1519:
1517:, p. 164.
1504:
1502:, p. 163.
1489:
1487:, p. 162.
1471:
1470:
1468:
1465:
1463:
1460:
1459:
1458:
1452:Barreled space
1447:
1444:
1442:-complete).
1439:
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991:
987:locally convex
978:
975:
974:
973:
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941:
932:linear map of
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720:
719:
707:
687:
678:linear map of
672:
660:
640:locally convex
627:
615:
612:
597:
593:
589:
586:
564:
559:
555:
534:
514:
494:
491:
488:
466:
461:
457:
434:
430:
426:
423:
413:
395:
391:
369:
358:locally convex
343:
339:
310:
306:
302:
299:
277:
272:
268:
247:
227:
207:
204:
201:
180:
176:
173:
168:
164:
159:
155:
133:
128:
124:
101:
97:
74:
70:
66:
63:
32:
18:locally convex
13:
10:
9:
6:
4:
3:
2:
2426:
2415:
2412:
2411:
2409:
2394:
2386:
2385:
2382:
2376:
2373:
2371:
2368:
2366:
2363:
2361:
2357:
2353:
2351:) convex
2350:
2347:
2345:
2342:
2340:
2336:
2334:
2331:
2329:
2326:
2324:
2323:Semi-complete
2321:
2319:
2316:
2314:
2311:
2309:
2305:
2302:
2300:
2296:
2294:
2291:
2289:
2286:
2284:
2281:
2279:
2276:
2274:
2271:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2244:
2241:
2239:
2238:Infrabarreled
2236:
2234:
2231:
2229:
2226:
2222:
2219:
2218:
2217:
2214:
2212:
2209:
2207:
2204:
2202:
2199:
2197:
2196:Distinguished
2194:
2192:
2189:
2187:
2184:
2182:
2179:
2177:
2174:
2172:
2168:
2164:
2162:
2159:
2157:
2153:
2149:
2147:
2144:
2142:
2139:
2137:
2134:
2133:
2131:
2129:Types of TVSs
2127:
2121:
2117:
2113:
2111:
2108:
2106:
2103:
2101:
2098:
2096:
2093:
2091:
2087:
2083:
2081:
2078:
2077:
2075:
2071:
2065:
2062:
2060:
2057:
2055:
2052:
2050:
2049:Prevalent/Shy
2047:
2045:
2042:
2040:
2039:Extreme point
2037:
2035:
2029:
2027:
2021:
2019:
2016:
2014:
2011:
2009:
2006:
2004:
2001:
1999:
1996:
1994:
1991:
1989:
1986:
1984:
1981:
1979:
1976:
1975:
1973:
1971:Types of sets
1969:
1963:
1960:
1958:
1955:
1953:
1950:
1948:
1945:
1941:
1938:
1936:
1933:
1931:
1928:
1927:
1926:
1923:
1921:
1918:
1914:
1913:Discontinuous
1911:
1909:
1906:
1904:
1901:
1899:
1896:
1894:
1891:
1889:
1886:
1884:
1881:
1880:
1879:
1876:
1872:
1869:
1868:
1867:
1864:
1863:
1861:
1857:
1851:
1848:
1844:
1841:
1840:
1839:
1836:
1833:
1830:
1828:
1824:
1821:
1819:
1816:
1814:
1811:
1809:
1806:
1804:
1801:
1800:
1798:
1796:
1792:
1786:
1783:
1781:
1778:
1776:
1773:
1771:
1770:Metrizability
1768:
1766:
1763:
1761:
1758:
1756:
1755:Fréchet space
1753:
1751:
1748:
1746:
1743:
1741:
1738:
1736:
1733:
1732:
1730:
1726:
1721:
1714:
1709:
1707:
1702:
1700:
1695:
1694:
1691:
1685:
1682:
1681:
1677:
1671:
1667:
1663:
1657:
1653:
1649:
1645:
1641:
1637:
1633:
1629:
1623:
1619:
1614:
1610:
1606:
1602:
1596:
1592:
1588:
1584:
1579:
1575:
1571:
1567:
1561:
1557:
1553:
1549:
1544:
1543:
1539:
1531:
1526:
1524:
1520:
1516:
1511:
1509:
1505:
1501:
1496:
1494:
1490:
1486:
1481:
1479:
1477:
1473:
1466:
1461:
1453:
1450:
1449:
1445:
1443:
1425:
1413:
1410:has a closed
1397:
1388:
1374:
1354:
1348:
1345:
1342:
1334:
1318:
1309:
1307:
1291:
1267:
1259:
1256:
1238:
1234:
1220:
1218:
1214:
1213:Fréchet space
1209:
1204:There exist B
1199:
1197:
1183:
1163:
1143:
1123:
1120:
1117:
1110:is closed in
1097:
1077:
1057:
1035:
1031:
1010:
1000:
990:
988:
984:
976:
959:
939:
931:
927:
923:
906:
902:
881:
874:
873:
872:
851:
845:
839:
816:
810:
790:
770:
762:
746:
740:
737:
734:
727:A linear map
726:
725:
724:
723:
722:
721:
705:
685:
677:
673:
658:
651:
650:
649:
646:
644:
641:
625:
613:
611:
591:
587:
584:
557:
553:
532:
512:
505:is closed in
492:
489:
486:
459:
455:
447:is closed in
428:
424:
421:
411:
409:
393:
389:
367:
359:
341:
337:
328:
324:
304:
300:
297:
270:
266:
245:
225:
218:is closed in
205:
202:
199:
178:
174:
171:
162:
157:
153:
126:
122:
95:
68:
64:
61:
53:
49:
47:
30:
22:
19:
2299:Polynomially
2228:Grothendieck
2221:tame Fréchet
2171:Bornological
2140:
2031:Linear cone
2023:Convex cone
1998:Banach disks
1940:Sesquilinear
1795:Main results
1785:Vector space
1740:Completeness
1735:Banach space
1647:
1617:
1582:
1547:
1540:Bibliography
1389:
1310:
1305:
1221:
1210:
1203:
1002:
992:
980:
870:
760:
647:
617:
381:
326:
325:
51:
45:
44:
15:
2293:Quasinormed
2206:FK-AK space
2100:Linear span
2095:Convex hull
2080:Affine hull
1883:Almost open
1823:Hahn–Banach
1335:surjection
1333:nearly open
930:nearly open
761:nearly open
676:nearly open
192:) whenever
2333:Stereotype
2191:(DF)-space
2186:Convenient
1925:Functional
1893:Continuous
1878:Linear map
1818:F. Riesz's
1760:Linear map
1467:References
1412:hyperplane
977:Properties
926:biunivocal
921:-complete.
759:is called
638:will be a
52:Ptak space
2349:Uniformly
2308:Reflexive
2156:Barrelled
2152:Countably
2064:Symmetric
1962:Transpose
1670:840278135
1636:144216834
1390:If a TVS
1352:→
1255:Hausdorff
1217:reflexive
1121:×
744:→
596:′
588:⊆
563:′
558:σ
490:∩
479:whenever
465:′
460:σ
433:′
425:⊆
414:subspace
410:if every
408:-complete
309:′
301:⊆
276:′
271:σ
203:∩
167:′
154:σ
132:′
127:σ
100:′
73:′
65:⊆
48:-complete
2408:Category
2393:Category
2344:Strictly
2318:Schwartz
2258:LF-space
2253:LB-space
2211:FK-space
2181:Complete
2161:BK-space
2086:Relative
2033:(subset)
2025:(subset)
1952:Seminorm
1935:Bilinear
1446:See also
1258:quotient
983:complete
645:(TVS).
2358:)
2306:)
2248:K-space
2233:Hilbert
2216:Fréchet
2201:F-space
2176:Brauner
2169:)
2154:)
2136:Asplund
2118:)
2088:)
2008:Bounded
1903:Compact
1888:Bounded
1825: (
1609:8588370
1574:4493665
2370:Webbed
2356:Quasi-
2278:Montel
2268:Mackey
2167:Ultra-
2146:Banach
2054:Radial
2018:Convex
1988:Affine
1930:Linear
1898:Closed
1722:(TVSs)
1668:
1658:
1634:
1624:
1607:
1597:
1572:
1562:
1280:is a B
1211:Every
525:(when
238:(when
114:(i.e.
23:(TVS)
2328:Smith
2313:Riesz
2304:Semi-
2116:Quasi
2110:Polar
1462:Notes
1304:is a
1136:. If
412:dense
50:or a
1947:Norm
1871:form
1859:Maps
1666:OCLC
1656:ISBN
1632:OCLC
1622:ISBN
1605:OCLC
1595:ISBN
1570:OCLC
1560:ISBN
1003:Let
360:TVS
1652:GTM
1311:If
894:is
380:is
146:or
43:is
2410::
1664:.
1650:.
1630:.
1603:.
1593:.
1585:.
1568:.
1558:.
1550:.
1522:^
1507:^
1492:^
1475:^
928:,
803:,
610:.
323:.
16:A
2354:(
2339:B
2337:(
2297:(
2165:(
2150:(
2114:(
2084:(
1834:)
1712:e
1705:t
1698:v
1672:.
1638:.
1611:.
1576:.
1440:r
1426:X
1416:r
1398:X
1375:X
1355:X
1349:P
1346::
1343:u
1319:X
1306:B
1292:X
1282:r
1268:X
1239:r
1235:B
1224:r
1206:r
1184:u
1164:X
1144:u
1124:Y
1118:X
1098:u
1078:Y
1058:X
1036:r
1032:B
1011:u
960:Y
940:X
907:r
903:B
882:X
852:.
849:)
846:X
843:(
840:u
820:)
817:U
814:(
811:u
791:X
771:U
747:Y
741:X
738::
735:u
706:Y
686:X
659:X
626:X
592:X
585:A
554:X
533:A
513:A
493:A
487:Q
456:X
429:X
422:Q
394:r
390:B
368:X
342:r
338:B
327:B
305:X
298:A
267:X
246:A
226:A
206:A
200:Q
179:)
175:X
172:,
163:X
158:(
123:X
96:X
69:X
62:Q
46:B
31:X
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