$ω^ω$-Base and infinite-dimensional compact sets in locally convex spaces

A locally convex space (lcs) $E$ is said to have an $ω^ω$-base if $E$ has a neighborhood base $\{U_α:α\inω^ω\}$ at zero such that $U_β\subseteq U_α$ for all $α\leqβ$. The class of lcs with an $ω^ω$-base is large, among others contains all $(LM)$-spaces (hence $(LF)$-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $D'(Ω)$). A remarkable result of Cascales-Orihuela states that every compact set in a lcs with an $ω^ω$-base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $ω^ω$-base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional space $φ$ endowed with the finest locally convex topology has an $ω^ω$-base but contains no infinite-dimensional compact subsets. It turns out that $φ$ 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.

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