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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.

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