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$K$-analytic space; web space; quasi-Suslin space
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 $\{A_{\alpha }\:\alpha \in \Bbb N^{\Bbb N}\}$ consisting of (relatively countably) compact sets such that $A_{\alpha }\subseteq A_{\beta }$ for $\alpha \leq \beta $. Some applications and two open questions are provided.
[1] Cascales, B.: On $K$-analytic locally convex spaces. Arch. Math. 49 (1987), 232-244. DOI 10.1007/BF01271663 | MR 0906738 | Zbl 0617.46014
[2] Cascales, B., Orihuela, J.: On compactness in locally convex spaces. Math. Z. 195 (1987), 365-381. DOI 10.1007/BF01161762 | MR 0895307 | Zbl 0604.46011
[3] 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). MR 0348724
[4] 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. MR 1270627
[5] Dierolf, P., Dierolf, S., Drewnowski, L.: Remarks and examples concerning unordered Baire-like and ultrabarrelled spaces. Colloq. Math. 39 (1978), 109-116. MR 0507270 | Zbl 0386.46008
[6] Drewnowski, L.: Resolutions of topological linear spaces and continuity of linear maps. J. Math. Anal. Appl. 335 (2007), 1177-1194. DOI 10.1016/j.jmaa.2007.02.032 | MR 2346899 | Zbl 1133.46002
[7] 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 Liège 1982 (unpublished Math.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).
[8] Drewnowski, L., Labuda, I.: Sequence $F$-spaces of $L_0$-type over submeasures of $\Bbb N$. (to appear).
[9] Kąkol, J., Pellicer, M. López: Compact coverings for Baire locally convex spaces. J. Math. Anal. Appl. 332 (2007), 965-974. DOI 10.1016/j.jmaa.2006.10.045 | MR 2324313
[10] Kelley, J. L., al., I. Namioka et: Linear Topological Spaces. Van Nostrand London (1963). MR 0166578 | Zbl 0115.09902
[11] Kōmura, Y.: On linear topological spaces. Kumamoto J. Sci., Ser. A 5 (1962), 148-157. MR 0151817
[12] Nakamura, M.: On quasi-Suslin space and closed graph theorem. Proc. Japan Acad. 46 (1970), 514-517. MR 0282325
[13] Nakamura, M.: On closed graph theorem. Proc. Japan Acad. 46 (1970), 846-849. MR 0291757 | Zbl 0223.46008
[14] Carreras, P. Perez, Bonet, J.: Barrelled Locally Convex Spaces, Vol. 131. North Holland Amsterdam (1987). MR 0880207
[15] 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).
[16] Talagrand, M.: Espaces de Banach faiblement $K$-analytiques. Ann. Math. 110 (1979), 407-438. DOI 10.2307/1971232 | MR 0554378
[17] Tkachuk, V. V.: A space $C_p(X) $ is dominated by irrationals if and only if it is $K$-analytic. Acta Math. Hungar. 107 (2005), 253-265. DOI 10.1007/s10474-005-0194-y | MR 2150789 | Zbl 1081.54012
[18] Valdivia, M.: Topics in Locally Convex Spaces. North-Holland Amsterdam (1982). MR 0671092 | Zbl 0489.46001
[19] Valdivia, M.: Quasi-LB-spaces. J. Lond. Math. Soc. 35 (1987), 149-168. MR 0871772 | Zbl 0625.46006
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