ωω-Base and infinite-dimensional compact sets in locally convex spaces

A locally convex space (lcs) E is said to have an omega(omega)-base if E has a neighborhood base (U-alpha : alpha is an element of omega(omega)} at zero such that U-beta subset of U-alpha for all alpha <= 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 Frdchet lcs (hence spaces of distributions D'(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 phi endowed with the finest locally convex topology has an omega(omega)-base but contains no infinite-dimensional compact subsets. It turns out that phi 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.