Function spaces of completely metrizable spaces

Baars, Jan; de Groot, Joost; Pelant, Jan

doi:10.1090/S0002-9947-1993-1160154-X

Function spaces of completely metrizable spaces
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by Jan Baars, Joost de Groot and Jan Pelant PDF

Trans. Amer. Math. Soc. 340 (1993), 871-883 Request permission

Abstract:

Let $X$ and $Y$ be metric spaces and let $\phi :{C_p}(X) \to {C_p}(Y)$ (resp. $\phi :C_p^\ast (X) \to C_p^\ast (Y)$) 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. $l_p^\ast$-equivalence) in the class of all metric spaces. This solves Problem 35 in [2] (raised by Arhangel’skiĭ).

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