Bounded sets structure of 𝑪𝒑 (𝑿) and quasi-(𝑫𝑭)-spaces

Abstract

For wide classes of locally convex spaces, in particular, for the space 𝐶𝑝(𝑋) of continuous real-valued functions on a Tychonoff space 𝑋 equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck’s theory of (𝐷𝐹)-spaces have led us to introduce quasi-(𝐷𝐹)-spaces, a class of locally convex spaces containing (𝐷𝐹)-spaces that preserves subspaces, countable direct sums and countable products. Regular (𝐿𝑀)-spaces as well as their strong duals are quasi- (𝐷𝐹)-spaces. Hence the space of distributions 𝐷′(Ω) provides a concrete example of a quasi-(𝐷𝐹)-space not being a (𝐷𝐹)-space. We show that 𝐶𝑝(𝑋) has a fundamental bounded resolution if and only if 𝐶𝑝(𝑋) is a quasi-(𝐷𝐹)-space if and only if the strong dual of 𝐶𝑝(𝑋) is a quasi-(𝐷𝐹)-space if and only if 𝑋 is countable. If 𝑋 is metrizable, then 𝐶𝑘(𝑋) is a quasi-(𝐷𝐹)-space if and only if 𝑋 is a 𝜎-compact Polish space.