# 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

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topFerrando, 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

top- Cascales, B., 10.1007/BF01271663, Arch. Math. 49 (1987), 232-244. (1987) Zbl0617.46014MR0906738DOI10.1007/BF01271663
- Cascales, B., Orihuela, J., 10.1007/BF01161762, Math. Z. 195 (1987), 365-381. (1987) Zbl0604.46011MR0895307DOI10.1007/BF01161762
- Christensen, J. P. R., Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, Vol. 10, North Holland Amsterdam (1974). (1974) MR0348724
- Comfort, W. W., Remus, D., Compact groups of Ulam-measurable cardinality: Partial converse theorems of Arkhangel'skii and Varopoulos, Math. Jap. 39 (1994), 203-210. (1994) MR1270627
- Dierolf, P., Dierolf, S., Drewnowski, L., Remarks and examples concerning unordered Baire-like and ultrabarrelled spaces, Colloq. Math. 39 (1978), 109-116. (1978) Zbl0386.46008MR0507270
- Drewnowski, L., 10.1016/j.jmaa.2007.02.032, J. Math. Anal. Appl. 335 (2007), 1177-1194. (2007) Zbl1133.46002MR2346899DOI10.1016/j.jmaa.2007.02.032
- Drewnowski, L., The dimension and codimension of analytic subspaces in topological vector spaces, with applications to the constructions of some pathological topological vector spaces. Liège 1982 (unpublished Math. talk), .
- Drewnowski, L., Labuda, I., Sequence $F$-spaces of ${L}_{0}$-type over submeasures of $\mathbb{N}$, (to appear).
- Kąkol, J., Pellicer, M. López, 10.1016/j.jmaa.2006.10.045, J. Math. Anal. Appl. 332 (2007), 965-974. (2007) MR2324313DOI10.1016/j.jmaa.2006.10.045
- Kelley, J. L., al., I. Namioka et, Linear Topological Spaces, Van Nostrand London (1963). (1963) Zbl0115.09902MR0166578
- Kōmura, Y., On linear topological spaces, Kumamoto J. Sci., Ser. A 5 (1962), 148-157. (1962) MR0151817
- Nakamura, M., On quasi-Suslin space and closed graph theorem, Proc. Japan Acad. 46 (1970), 514-517. (1970) MR0282325
- Nakamura, M., On closed graph theorem, Proc. Japan Acad. 46 (1970), 846-849. (1970) Zbl0223.46008MR0291757
- Carreras, P. Perez, Bonet, J., Barrelled Locally Convex Spaces, Vol. 131, North Holland Amsterdam (1987). (1987) MR0880207
- Rogers, C. A., Jayne, J. E., Dellacherie, C., Topsøe, F., Hoffman-Jørgensen, J., Martin, D. A., Kechris, A. S., Stone, A. H., Analytic Sets, Academic Press London (1980). (1980)
- Talagrand, M., 10.2307/1971232, Ann. Math. 110 (1979), 407-438. (1979) MR0554378DOI10.2307/1971232
- Tkachuk, V. V., 10.1007/s10474-005-0194-y, Acta Math. Hungar. 107 (2005), 253-265. (2005) Zbl1081.54012MR2150789DOI10.1007/s10474-005-0194-y
- Valdivia, M., Topics in Locally Convex Spaces, North-Holland Amsterdam (1982). (1982) Zbl0489.46001MR0671092
- Valdivia, M., Quasi-LB-spaces, J. Lond. Math. Soc. 35 (1987), 149-168. (1987) Zbl0625.46006MR0871772

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