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 an 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 vector space \(\varphi \) endowed with the finest locally convex topology has an \(\omega ^\omega \)-base but contains no infinite-dimensional compact subsets. It turns out that \(\varphi \) 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.

甲局部凸空间（LCS）é据说有一个\（\欧米加^ {\欧米加} \）碱基如果Ë具有邻域基\（\ {U _ {\阿尔法}：\阿尔法\在\欧米加^ \ omega \} \）为零，因此对于所有\（\ alpha \ le \ beta \）都是\（U _ {\ beta} \ subseteq U _ {\ alpha} \）。具有\（\ omega ^ {\ omega} \） -base的lcs类别很大，其中包含所有（LM）-空间（因此（LF）-空间），杰出的Fréchetlcs的强对偶（因此空间为分布\（D ^ {\ prime}（\ Omega）\））。Cascales-Orihuela的一项非凡结果表明，在具有\（\ omega ^ {\ omega} \）的lcs中的每个紧凑集-base是可计量的。我们的主要结果表明，每个具有\（\ omega ^ {\ omega} \）-基的不可数维lcs都包含一个无限维的可量化紧致子集。另一方面，具有最佳局部凸拓扑的可数维向量空间\（\ varphi \）具有\（\ omega ^ \ omega \）-基，但不包含无限维紧凑子集。事实证明，\（\ varphi \）是唯一的无限维局部凸空间，它是不包含无限维紧凑子集的\（k _ {\ mathbb {R}} \） -空间。提供了对空格\（C_ {p}（X）\）的应用程序。