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.
@article{118628,
author = {Jan Baars},
title = {A note on linear mappings between function spaces},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {34},
year = {1993},
pages = {711-715},
zbl = {0787.54017},
mrnumber = {1263800},
language = {en},
url = {http://dml.mathdoc.fr/item/118628}
}
Baars, Jan. A note on linear mappings between function spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) pp. 711-715. http://gdmltest.u-ga.fr/item/118628/
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