A revised closed graph theorem for quasi-Suslin spaces

Juan Carlos Ferrando; J. Kąkol; M. Lopez Pellicer

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 4, page 1115-1122
  • ISSN: 0011-4642

Abstract

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Some results about the continuity of special linear maps between F -spaces recently obtained by Drewnowski have motivated us to revise a closed graph theorem for quasi-Suslin spaces due to Valdivia. We extend Valdivia’s theorem by showing that a linear map with closed graph from a Baire tvs into a tvs admitting a relatively countably compact resolution is continuous. This also applies to extend a result of De Wilde and Sunyach. A topological space X is said to have a (relatively countably) compact resolution if X admits a covering { A α α ∈ ℕ ℕ } consisting of (relatively countably) compact sets such that A α ⊆ A β for α ≤ β . Some applications and two open questions are provided.

How to cite

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Ferrando, Juan Carlos, Kąkol, J., and Lopez Pellicer, M.. "A revised closed graph theorem for quasi-Suslin spaces." Czechoslovak Mathematical Journal 59.4 (2009): 1115-1122. <http://eudml.org/doc/37982>.

@article{Ferrando2009,
abstract = {Some results about the continuity of special linear maps between $F$-spaces recently obtained by Drewnowski have motivated us to revise a closed graph theorem for quasi-Suslin spaces due to Valdivia. We extend Valdivia’s theorem by showing that a linear map with closed graph from a Baire tvs into a tvs admitting a relatively countably compact resolution is continuous. This also applies to extend a result of De Wilde and Sunyach. A topological space $X$ is said to have a (relatively countably) compact resolution if $X$ admits a covering $\lbrace A_\{\alpha \}\:\alpha \in \mathbb \{N\}^\{\mathbb \{N\}\}\rbrace $ consisting of (relatively countably) compact sets such that $A_\{\alpha \}\subseteq A_\{\beta \}$ for $\alpha \le \beta $. Some applications and two open questions are provided.},
author = {Ferrando, Juan Carlos, Kąkol, J., Lopez Pellicer, M.},
journal = {Czechoslovak Mathematical Journal},
keywords = {$K$-analytic space; web space; quasi-Suslin space; -analytic space; webbed space; quasi-Suslin space},
language = {eng},
number = {4},
pages = {1115-1122},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A revised closed graph theorem for quasi-Suslin spaces},
url = {http://eudml.org/doc/37982},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Ferrando, Juan Carlos
AU - Kąkol, J.
AU - Lopez Pellicer, M.
TI - A revised closed graph theorem for quasi-Suslin spaces
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 4
SP - 1115
EP - 1122
AB - Some results about the continuity of special linear maps between $F$-spaces recently obtained by Drewnowski have motivated us to revise a closed graph theorem for quasi-Suslin spaces due to Valdivia. We extend Valdivia’s theorem by showing that a linear map with closed graph from a Baire tvs into a tvs admitting a relatively countably compact resolution is continuous. This also applies to extend a result of De Wilde and Sunyach. A topological space $X$ is said to have a (relatively countably) compact resolution if $X$ admits a covering $\lbrace A_{\alpha }\:\alpha \in \mathbb {N}^{\mathbb {N}}\rbrace $ consisting of (relatively countably) compact sets such that $A_{\alpha }\subseteq A_{\beta }$ for $\alpha \le \beta $. Some applications and two open questions are provided.
LA - eng
KW - $K$-analytic space; web space; quasi-Suslin space; -analytic space; webbed space; quasi-Suslin space
UR - http://eudml.org/doc/37982
ER -

References

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