${\omega }^{\omega }$-Base and infinite-dimensional compact sets in locally convex spaces

Abstract

A locally convex space (lcs) $E$ is said to have an ${\omega }^{\omega }$-base if $E$ has a neighborhood base $\{U_{\alpha }:\alpha \in {\omega }^{\omega }\}$ at zero such that $U_{\beta }\subseteq U_{\alpha }$ for all $\alpha \le \beta$. The class of lcs with an ${\omega }^{\omega }$-base is large, among others contains all $(LM)$-spaces (hence $(LF)$-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $D^{\prime }(\Omega )$). A remarkable result of Cascales-Orihuela states that every compact set in a lcs with an ${\omega }^{\omega }$-base is metrizable. Our main result shows that every uncountable-dimensional lcs with an ${\omega }^{\omega }$-base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional space $\phi$ endowed with the finest locally convex topology has an ${\omega }^{\omega }$-base but contains no infinite-dimensional compact subsets. It turns out that $\phi$ is a unique infinite-dimensional locally convex space which is a $k_{\mathbb{R}}$-space containing no infinite-dimensional compact subsets. Applications to spaces $C_p(X)$ are provided.