\(\omega^{\omega}\)-base and infinite-dimensional compact sets in locally convex spaces (Q2141406)

A locally convex Hausdorff space \(E\) is said to have an \(\mathbb{N}^{\mathbb{N}}\)-base if \(E\) has a basis \((U_{\alpha})\) of neighbourhoods of the origin such that \(U_{\beta} \subset U_{\alpha}\) if \(\alpha \leq \beta\) in \(\mathbb{N}^{\mathbb{N}}\). \textit{B. Cascales} and \textit{J. Orihuela} [Math. Z. 195, 365--381 (1987; Zbl 0604.46011)] showed that every compact set in a locally convex Hausdorff space with an \(\mathbb{N}^{\mathbb{N}}\)-base is metrizable. The authors prove the following results: (1) Every uncountably dimensional locally convex Hausdorff with an \(\mathbb{N}^{\mathbb{N}}\)-base contains an infinite dimensional metrizable compact subset. (2) A locally convex Hausdorff space \(E\) is topologically isomorphic to the the countable-dimensional vector space \(\varphi\) endowed with the finest locally convex topology if and only if \(E\) is a completely regular topological space containing no infinite-dimensional compact subsets, and if and only if \(E\) is bornological, has an \(\mathbb{N}^{\mathbb{N}}\)-base and contains no infinite-dimensional compact subset. Some applications to spaces of continuous functions on a completely regular topological space endowed with the pointwise topology are given.