# Article

 Title: A note on linear mappings between function spaces (English) Author: Baars, Jan Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 34 Issue: 4 Year: 1993 Pages: 711-715 . Category: math . Summary: Arhangel'ski\v{\i} proved that if $X$ and $Y$ are completely regular spaces such that ${C_p (X)}$ and ${C_p (Y)}$ are linearly homeomorphic, then $X$ is pseudocompact if and only if $Y$ is pseudocompact. In addition he proved the same result for compactness, $\sigma$-compactness and realcompactness. In this paper we prove that if $\phi : {C_p (X)} \rightarrow {C_p (X)}$ is a continuous linear surjection, then $Y$ is pseudocompact provided $X$ is and if $\phi$ is a continuous linear injection, then $X$ is pseudocompact provided $Y$ is. We also give examples that both statements do not hold for compactness, $\sigma$-compactness and realcompactness. (English) Keyword: function space Keyword: topology of pointwise convergence MSC: 54C35 MSC: 57N17 idZBL: Zbl 0787.54017 idMR: MR1263800 . Date available: 2009-01-08T18:07:38Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/118628 . Reference: [1] Arhangel'skiĭ A.V.: On linear homeomorphisms of function spaces.Soviet Math. Dokl. 25 (1982), 852-855. Reference: [2] Baars J., de Groot J.: On Topological and Linear Equivalence of Certain Function Spaces.CWI-tract 86, Centre for Mathematics and Computer Science, Amsterdam. Zbl 0755.54007, MR 1182148 Reference: [3] Baars J., de Groot J., Pelant J.: Function spaces of completely metrizable spaces.to appear in Trans. of the AMS. Zbl 0841.54012, MR 1182148 .

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