Baire-Type Properties of Topological Vector Spaces

Abstract

Abstract Burzyk, Kliś and Lipecki proved that every metrizable topological vector space (tvs) E with the property ( K ) is a Baire space. Ka̧kol and Sánchez Ruiz proved that every sequentially complete Fréchet–Urysohn locally convex space (lcs) is Baire. Being motivated by the property ( K ) and the notion of a Mackey null sequence we introduce a property ( MK ) which is strictly weaker than the property ( K ), and show that any locally complete lcs has the property ( MK ). We prove that any $$\kappa $$ κ -Fréchet–Urysohn tvs with the property ( MK ) is a Baire space; consequently, each locally complete $$\kappa $$ κ -Fréchet–Urysohn lcs is a Baire space. This generalizes both the aforementioned results. We construct a feral Baire space E with the property ( K ) and which is not $$\kappa $$ κ -Fréchet–Urysohn. Although a $$\kappa $$ κ -Fréchet–Urysohn lcs E can be not a Baire space, we show that E is always b -Baire-like in the sense of Ruess. Applications to spaces of Baire functions and $$C_k$$ C k -spaces are given.