## Jan Baars

*A note on linear mappings between function spaces *

Comment.Math.Univ.Carolinae 34,4 (1993) 711-715. **Abstract:**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 $\phi : {C_p (X)}\rightarrow {C_p (Y)}$ 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.

**Keywords:** function space, topology of pointwise convergence

**AMS Subject Classification:** 54C35, 57N17

PDF