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

Abstract

A locally convex space (lcs) E is said to have an ω ω -base if E has a neighborhood base {U α : α ∈ ω ω } at zero such that U β ⊆ U α for all α ≤ β. 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 R -space containing no infinite-dimensional compact subsets. Applications to spaces C p (X) are provided.