Intermediate vector topologies with an intrinsic hemicompactness criterion

Determine whether there exists a natural vector topology on the free vector space E(X) that is stronger than the free locally pseudoconvex topology L^{(0)}(X) and weaker than the free topological vector-space topology V(X), such that hemicompactness is determined by X but satisfies a criterion different from those for topologies between L_p(X) and L^{(0)}(X) and for V(X).

Background

The paper characterizes hemicompactness for several free topological vector spaces. For vector topologies between L_p(X) and L^{(0)}(X), hemicompactness is equivalent to X being countable with all compact subsets finite, whereas V(X) is hemicompact exactly when X is hemicompact.

The authors ask whether a meaningful intermediate topology, lying strictly above L^{(0)}(X) and below V(X), can yield a genuinely different criterion depending only on X. They note that 'natural' is not a strictly mathematical notion and therefore formulate a more formal version using prevarieties and varieties of topological vector spaces.