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 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.