Convex-compact sets and Banach discs

I. Monterde; Vicente Montesinos

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 3, page 773-780
  • ISSN: 0011-4642

Abstract

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Every relatively convex-compact convex subset of a locally convex space is contained in a Banach disc. Moreover, an upper bound for the class of sets which are contained in a Banach disc is presented. If the topological dual E ' of a locally convex space E is the σ ( E ' , E ) -closure of the union of countably many σ ( E ' , E ) -relatively countably compacts sets, then every weakly (relatively) convex-compact set is weakly (relatively) compact.

How to cite

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Monterde, I., and Montesinos, Vicente. "Convex-compact sets and Banach discs." Czechoslovak Mathematical Journal 59.3 (2009): 773-780. <http://eudml.org/doc/37957>.

@article{Monterde2009,
abstract = {Every relatively convex-compact convex subset of a locally convex space is contained in a Banach disc. Moreover, an upper bound for the class of sets which are contained in a Banach disc is presented. If the topological dual $E^\{\prime \}$ of a locally convex space $E$ is the $\sigma (E^\{\prime \},E)$-closure of the union of countably many $\sigma (E^\{\prime \},E)$-relatively countably compacts sets, then every weakly (relatively) convex-compact set is weakly (relatively) compact.},
author = {Monterde, I., Montesinos, Vicente},
journal = {Czechoslovak Mathematical Journal},
keywords = {weakly compact sets; convex-compact sets; Banach discs; weakly compact set; convex-compact set; Banach disc},
language = {eng},
number = {3},
pages = {773-780},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Convex-compact sets and Banach discs},
url = {http://eudml.org/doc/37957},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Monterde, I.
AU - Montesinos, Vicente
TI - Convex-compact sets and Banach discs
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 3
SP - 773
EP - 780
AB - Every relatively convex-compact convex subset of a locally convex space is contained in a Banach disc. Moreover, an upper bound for the class of sets which are contained in a Banach disc is presented. If the topological dual $E^{\prime }$ of a locally convex space $E$ is the $\sigma (E^{\prime },E)$-closure of the union of countably many $\sigma (E^{\prime },E)$-relatively countably compacts sets, then every weakly (relatively) convex-compact set is weakly (relatively) compact.
LA - eng
KW - weakly compact sets; convex-compact sets; Banach discs; weakly compact set; convex-compact set; Banach disc
UR - http://eudml.org/doc/37957
ER -

References

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  1. Day, M. M., Normed Linear Spaces, Spriger-Verlag (1973). (1973) Zbl0268.46013MR0344849
  2. Floret, K., 10.1007/BFb0091483, Lecture Notes in Math., Springer-Verlag 801 (1980). (1980) Zbl0437.46006MR0576235DOI10.1007/BFb0091483
  3. Grothendieck, A., 10.2307/2372076, Amer. J. Math. 74 (1952), 168-186. (1952) Zbl0046.11702MR0047313DOI10.2307/2372076
  4. Köthe, G., Topological Vector Spaces I, Springer-Verlag (1969). (1969) MR0248498
  5. Pták, V., A combinatorial lemma on the existence of convex means and its applications to weak compactness, Proc. Symp. Pure Math. VII (Convexity 1963) 437-450. MR0161128

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