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

Abstract

A locally convex space (lcs) E is said to have an $$\omega ^{\omega }$$ ω ω -base if E has a neighborhood base $$\{U_{\alpha }:\alpha \in \omega ^\omega \}$$ { U α : α ∈ ω ω } at zero such that $$U_{\beta }\subseteq U_{\alpha }$$ U β ⊆ U α for all $$\alpha \le \beta $$ α ≤ β . The class of lcs with an $$\omega ^{\omega }$$ ω ω -base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $$D^{\prime }(\Omega )$$ D ′ ( Ω ) ). A remarkable result of Cascales-Orihuela states that every compact set in an lcs with an $$\omega ^{\omega }$$ ω ω -base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $$\omega ^{\omega }$$ ω ω -base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional vector space $$\varphi $$ φ endowed with the finest locally convex topology has an $$\omega ^\omega $$ ω ω -base but contains no infinite-dimensional compact subsets. It turns out that $$\varphi $$ φ is a unique infinite-dimensional locally convex space which is a $$k_{\mathbb {R}}$$ k R -space containing no infinite-dimensional compact subsets. Applications to spaces $$C_{p}(X)$$ C p ( X ) are provided.

Sub-posets in $ω^ω$ and the Strong Pytkeev$^\ast$ Property

“…Then we show that each uncountably-dimensional locally convex space (lcs) E with a Pbase contains an infinite-dimensional metrizable compact subspace if P is a directed set equipped with a second-countable topology in which every convergent sequence in P is bounded. This extended Theorem 1.2 in [3]. Examples in function spaces are given.…”

“…Motivated by these results, it is natural to investigate the class of lcs in which every infinite-dimensional subspace contains an infinite-dimensional compact metrizable subset. In [3], one of the main results is that every uncountably-dimensional lcs with an ω ω -base contains an infinitedimensional compact metrizable subset. Hence C k (X) satisfies this property if X is a infinite continuous image of a Polish space under a perfect mapping.…”

“…A lcs E is said to have (κ, λ)-tall bornology if every κ-sized subset A contains a λ-sized bounded subset. It is known (see [3]) that any lcs which is (κ, λ) pequicovergent has (κ, λ)-tall bornology. The authors in [3] also prove that any topological space with countable cs • -network at a point x is (ω 1 , ω) p -equiconvergent at x.…”

“…It is known (see [3]) that any lcs which is (κ, λ) pequicovergent has (κ, λ)-tall bornology. The authors in [3] also prove that any topological space with countable cs • -network at a point x is (ω 1 , ω) p -equiconvergent at x. A family N of subsets of X is said to be a cs • -network at a point x ∈ X if for any neighborhood O x of x and any sequence {x n : n ∈ ω} converging to x there exists N ∈ N such that N ⊂ O x and N contains some point x n in the sequence.…”

“…It is proved [3] that for any lcs E each compact subset of E has finite topological dimension if and only if each bounded linearly independent subset of E is finite. We use this result to prove the following theorem.…”