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Function spaces of completely metrizable spaces. (English) Zbl 0841.54012
Summary: Let \(X\) and \(Y\) be metric spaces and let \(\varphi : C_p (X) \to C_p (Y)\) (resp. \(\varphi : C_p^* (X) \to C_p^* (Y))\) be a continuous linear surjection. We prove that \(Y\) is completely metrizable whenever \(X\) is. As a corollary we obtain that complete metrizability is preserved by \(l_p\)- (resp. \(l^*_p\)-)equivalence in the class of all metric spaces. This solves Problem 35 in [Problems in \(C_p\)-theory, in ‘Open problems in topology’ (1990; Zbl 0718.54001)] (raised by Arkhangel’skij).

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