Absolutely convex sets in barrelled spaces

Valdivia, Manuel

doi:10.5802/aif.368

Valdivia, Manuel

Annales de l'Institut Fourier, Tome 21 (1971) no. 2, pp. 3-13.

Résumé
Abstract

Étant donné une suite croissante ${A_{n}}$ d’ensembles absolument convexes dans un espace tonnelé $E$ , de manière que $⋃_{n = 1}^{\infty} A_{n} = E$ , on déduit quelques propriétés de $E$ à partir des propriétés des ensembles de ${A_{n}}$ . On démontre que dans un espace tonnelé quelconque, tout sous-espace de codimension infinie dénombrable est tonnelé.

If ${A_{n}}$ is an increasing sequence of absolutely convex sets, in a barrelled space $E$ , such that $⋃_{n = 1}^{\infty} A_{n} = E$ , it is deduced some properties of $E$ from the properties of the sets of ${A_{n}}$ . It is shown that in a barrelled space any subspace of infinite countable codimension, is barrelled.

MR Zbl | 5 citations dans Numdam

DOI : 10.5802/aif.368

@article{AIF_1971__21_2_3_0,
     author = {Valdivia, Manuel},
     title = {Absolutely convex sets in barrelled spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {3--13},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {21},
     number = {2},
     year = {1971},
     doi = {10.5802/aif.368},
     mrnumber = {48 #11968},
     zbl = {0205.40904},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.368/}
}

TY  - JOUR
AU  - Valdivia, Manuel
TI  - Absolutely convex sets in barrelled spaces
JO  - Annales de l'Institut Fourier
PY  - 1971
SP  - 3
EP  - 13
VL  - 21
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - https://www.numdam.org/articles/10.5802/aif.368/
DO  - 10.5802/aif.368
LA  - en
ID  - AIF_1971__21_2_3_0
ER  -

%0 Journal Article
%A Valdivia, Manuel
%T Absolutely convex sets in barrelled spaces
%J Annales de l'Institut Fourier
%D 1971
%P 3-13
%V 21
%N 2
%I Institut Fourier
%C Grenoble
%U https://www.numdam.org/articles/10.5802/aif.368/
%R 10.5802/aif.368
%G en
%F AIF_1971__21_2_3_0

Valdivia, Manuel. Absolutely convex sets in barrelled spaces. Annales de l'Institut Fourier, Tome 21 (1971) no. 2, pp. 3-13. doi : 10.5802/aif.368. https://www.numdam.org/articles/10.5802/aif.368/

Bibliographie
Cité par

[1] I. Amemiya und Y. Komura, Uber nicht-vollständige Montelräume. Math. Annalen. 177, 273-277, (1968). | MR | Zbl

[2] M. De Wilde, Sur les sous-espaces de codimension finie d'un espace linéaire à semi-normes, To appear in Bull. Soc. R. Sciences, Liége. | Zbl

[3] J. Dieudonné, Sur les propriétés de permanence de certains espaces vectoriels topologiques, Ann. Soc. Pol. Math. 25, 50-55, (1952). | MR | Zbl

[4] J. Horváth, Topological Vector Spaces and Distributions. I. Massachussets (1966). | MR | Zbl

[5] G. Köthe, Die Bilräume abgeschlossener Operatoren, J. für die reine und And. Math., 232, 110-111, (1968). | Zbl

Kąkol, Jerzy; Kubiś, Wiesław; López-Pellicer, Manuel; Sobota, Damian Elementary Facts about Baire and Baire-Type Spaces, Descriptive Topology in Selected Topics of Functional Analysis, Volume 24 (2025), p. 15 | DOI:10.1007/978-3-031-76062-4_2
Kališnik, J. Reflexivity of the Space of Transversal Distributions, The Journal of Geometric Analysis, Volume 33 (2023) no. 10 | DOI:10.1007/s12220-023-01390-y
Kąkol, Jerzy; Saxon, Stephen A. The quotient/codimension problems, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, Volume 113 (2019) no. 2, p. 1429 | DOI:10.1007/s13398-018-0556-2
Saxon, Stephen A. Mackey hyperplanes/enlargements for Tweddle’s space, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, Volume 108 (2014) no. 2, p. 1035 | DOI:10.1007/s13398-013-0159-x
Valdivia, Manuel The space H(Ω,(zj)) of holomorphic functions, Journal of Mathematical Analysis and Applications, Volume 337 (2008) no. 2, p. 821 | DOI:10.1016/j.jmaa.2007.03.070
Kąkol, Jerzy; Saxon, Stephen A. The Semi-Baire-like Three-Space Problems, Advances in Applied Mathematics, Volume 25 (2000) no. 1, p. 57 | DOI:10.1006/aama.1999.0690
Saxon, Stephen; Ruiz, L. Dual local completeness, Proceedings of the American Mathematical Society, Volume 125 (1997) no. 4, p. 1063 | DOI:10.1090/s0002-9939-97-03864-1
Burke, Maxim R.; Todorcevic, Stevo Bounded sets in topological vector spaces, Mathematische Annalen, Volume 305 (1996) no. 1, p. 103 | DOI:10.1007/bf01444213
Ferrer, J.; López-Pellicer, M. Algebras of sets generating non-barrelled spaces, Periodica Mathematica Hungarica, Volume 29 (1994) no. 3, p. 207 | DOI:10.1007/bf01875850
Narayanaswami, P. P. The separable quotient problem for barrelled spaces, Functional Analysis and Related Topics, 1991, Volume 1540 (1993), p. 289 | DOI:10.1007/bfb0085487
Todd, Aaron R. Beside and Between Baire and Barrelled or Linear Variations on Category, Annals of the New York Academy of Sciences, Volume 607 (1990) no. 1, p. 128 | DOI:10.1111/j.1749-6632.1990.tb22752.x
Ferrando, J. C.; L�pez-Pellicer, M. Strong barrelledness properties inl 0 ? (X A) and bounded finite additive measures, Mathematische Annalen, Volume 287 (1990) no. 1, p. 727 | DOI:10.1007/bf01446926
Ferrando, J. C.; López-Pellicer, M. Quasi-suprabarrelled spaces, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, Volume 46 (1989) no. 1, p. 137 | DOI:10.1017/s1446788700030433
Gim�nez, I.; L�pez, J. A. On weak and Schauder ?-bases in topological vector spaces, Archiv der Mathematik, Volume 51 (1988) no. 6, p. 561 | DOI:10.1007/bf01261978
References, Barrelled Locally Convex Spaces, Volume 131 (1987), p. 484 | DOI:10.1016/s0304-0208(08)71679-7
Valdivia, M. (F)-spaces and strict (LF)-spaces, Mathematische Zeitschrift, Volume 195 (1987) no. 3, p. 345 | DOI:10.1007/bf01161761
Chapter I–Mixed Topologies, Saks Spaces and Applications to Functional Analysis, Volume 139 (1987), p. 1 | DOI:10.1016/s0304-0208(08)72317-x
Orihuela, Jos� On the equivalence of weak and Schauder bases, Archiv der Mathematik, Volume 46 (1986) no. 5, p. 447 | DOI:10.1007/bf01210785
Narayanaswami, P.P. Classification of (LF)-Spaces by Some Baire-Like Covering Properties, Complex Analysis, Functional Analysis and Approximation Theory, Proceedings of the Conference on Complex Analysis and Approximation Theory Universidade Estadual de Campinas, Volume 125 (1986), p. 247 | DOI:10.1016/s0304-0208(08)72172-8
Valdivia, M. On Certain Metrizable Locally Convex Spaces, Complex Analysis, Functional Analysis and Approximation Theory, Proceedings of the Conference on Complex Analysis and Approximation Theory Universidade Estadual de Campinas, Volume 125 (1986), p. 287 | DOI:10.1016/s0304-0208(08)72176-5
Narayanaswami, P. P.; Saxon, Stephen A. (LF)-spaces, quasi-baire spaces and the strongest locally convex topology, Mathematische Annalen, Volume 274 (1986) no. 4, p. 627 | DOI:10.1007/bf01458598
Kąkol, Jerzy On Subspaces ofb-Spaces in the Sense of Noureddine, Mathematische Nachrichten, Volume 120 (1985) no. 1, p. 43 | DOI:10.1002/mana.19851200105
Valdivia, Manuel A Property of Fréchet Spaces, Functional Analysis, Holomorphy and Approximation Theory II, Volume 86 (1984), p. 469 | DOI:10.1016/s0304-0208(08)70841-7
Kąkol, Jerzy On subspaces and products of some topological linear spaces, Archiv der Mathematik, Volume 40 (1983) no. 1, p. 355 | DOI:10.1007/bf01192795
Carreras, Pedro Pérez; Bonet, José Remarks and examples concerning suprabarreled and totally barreled spaces, Archiv der Mathematik, Volume 39 (1982) no. 4, p. 340 | DOI:10.1007/bf01899442
Mukherjee, T. K.; Summers, W. H. Stability and closed graph theorems in classes of bornological spaces, Glasgow Mathematical Journal, Volume 23 (1982) no. 2, p. 151 | DOI:10.1017/s0017089500004924
References, Topics in Locally Convex Spaces, Volume 67 (1982), p. 501 | DOI:10.1016/s0304-0208(08)70400-6
Valdivia, Manuel On suprabarrelled spaces, Functional Analysis, Holomorphy, and Approximation Theory, Volume 843 (1981), p. 572 | DOI:10.1007/bfb0089291
Valdivia, Manuel; P�rez Carreras, Pedro On totally barreled spaces, Mathematische Zeitschrift, Volume 178 (1981) no. 2, p. 263 | DOI:10.1007/bf01262043
Moltó, Aníbal On uniform boundedness properties in exhausting additive set function spaces, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 90 (1981) no. 1-2, p. 175 | DOI:10.1017/s0308210500015419
Webb, J H SUBSPACES OF BARRELLED SPACES, Quaestiones Mathematicae, Volume 4 (1981) no. 4, p. 323 | DOI:10.1080/16073606.1981.9632252
De Wilde, Marc; Tsirulnikov, Bella Barrelled spaces with a B-complete completion, manuscripta mathematica, Volume 33 (1981) no. 3-4, p. 411 | DOI:10.1007/bf01798237
Robertson, W.J.; Tweddle, I.; Yeomans, F.E. On the stability of barrelled topologies, III, Bulletin of the Australian Mathematical Society, Volume 22 (1980) no. 1, p. 99 | DOI:10.1017/s0004972700006377
Saiflu, H.; Tweddle, I. From B-completeness to countable codimensional subspaces via the closed graph theorem, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 86 (1980) no. 1-2, p. 107 | DOI:10.1017/s030821050001204x
Robertson, W.J.; Yeomans, F.E. On the stability of barrelled topologies, I, Bulletin of the Australian Mathematical Society, Volume 20 (1979) no. 3, p. 385 | DOI:10.1017/s0004972700011096
Marquina, A.; Serna, J. M. Sanz Barrelledness conditions onC 0 (E), Archiv der Mathematik, Volume 31 (1978) no. 1, p. 589 | DOI:10.1007/bf01226496
Spaces with an absorbing sequence, Topological Vector Spaces, Volume 639 (1978), p. 84 | DOI:10.1007/bfb0068531
Husain, T.; Wong, Yau-Chuen On various types of barrelledness and the hereditary property of (DF)-spaces, Glasgow Mathematical Journal, Volume 17 (1976) no. 2, p. 134 | DOI:10.1017/s001708950000286x
Popoola, J. O.; Tweddle, I. On the closed graph theorem, Glasgow Mathematical Journal, Volume 17 (1976) no. 2, p. 89 | DOI:10.1017/s0017089500002780
Valdivia, M. On countable locally convex direct sums, Archiv der Mathematik, Volume 26 (1975) no. 1, p. 407 | DOI:10.1007/bf01229759
Dierolf, Susanne �ber assoziierte lineare und lokalkonvexe Topologien, Manuscripta Mathematica, Volume 16 (1975) no. 1, p. 27 | DOI:10.1007/bf01169061
Muni, Giuseppe Spazi a-tonnelés, Annali di Matematica Pura ed Applicata, Volume 99 (1974) no. 1, p. 191 | DOI:10.1007/bf02413725
Eberhardt, Volker �ber einen Graphensatz f�r Abbildungen mit normiertem Zielraum, Manuscripta Mathematica, Volume 12 (1974) no. 1, p. 47 | DOI:10.1007/bf01166233
Lurje, Pawel Tonneliertheit in lokalkonvexen Vektorgruppen, Manuscripta Mathematica, Volume 14 (1974) no. 2, p. 107 | DOI:10.1007/bf01171437
Ruess, W. A Grothendieck representation for the completion of cones of continuous seminorms, Mathematische Annalen, Volume 208 (1974) no. 1, p. 71 | DOI:10.1007/bf01432521
Mukherjee, T. K. Hereditary completeness and quasi-reflexivity, Proceedings of the American Mathematical Society, Volume 46 (1974) no. 2, p. 285 | DOI:10.1090/s0002-9939-1974-0352944-1
Valdivia, Manuel On a class of quasi-barrelled spaces, Mathematische Annalen, Volume 202 (1973) no. 4, p. 295 | DOI:10.1007/bf01433459
Todd, Aaron R.; Saxon, Stephen A. A property of locally convex Baire spaces, Mathematische Annalen, Volume 206 (1973) no. 1, p. 23 | DOI:10.1007/bf01431526
Webb, J. H. Countable-codimensional subspaces of locally convex spaces, Proceedings of the Edinburgh Mathematical Society, Volume 18 (1973) no. 3, p. 167 | DOI:10.1017/s0013091500009895
De Wilde, M. Various types of barrelledness and increasing sequences of balanced and convex sets in locally convex spaces, Summer School on Topological Vector Spaces, Volume 331 (1973), p. 211 | DOI:10.1007/bfb0068232
Horváth, John Locally convex spaces, Summer School on Topological Vector Spaces, Volume 331 (1973), p. 41 | DOI:10.1007/bfb0068227
Saxon, Stephen A. Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology, Mathematische Annalen, Volume 197 (1972) no. 2, p. 87 | DOI:10.1007/bf01419586
Roelcke, W. On the finest locally convex topology agreeing with a given topology on a sequence of absolutely convex sets, Mathematische Annalen, Volume 198 (1972) no. 1, p. 57 | DOI:10.1007/bf01420498
Valdivia, Manuel OnD? spaces, Mathematische Annalen, Volume 191 (1971) no. 1, p. 38 | DOI:10.1007/bf01433469

Cité par 54 documents. Sources : Crossref