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A note on linear mappings between function spaces. (English) Zbl 0787.54017
Summary: A. V. Arkhangel’skij [Sov. Math., Dokl. 25, 852-855 (1982); translation from Dokl. Akad. Nauk SSSR 264, 1289-1292 (1982; Zbl 0522.54015)] 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 \(\phi: C_ p(X)\to C_ p(Y)\) 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.

54C35 Function spaces in general topology
57N17 Topology of topological vector spaces
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