On the Set of Locally Convex Topologies Compatible with a Given Topology on a Vector Space: Cardinality Aspects

Abstract

For a topological vector space (X, τ), we consider the family LCT(X, τ) of all locally convex topologies defined on X, which give rise to the same continuous linear functionals as the original topology τ. We prove that for an infinite-dimensional reflexive Banach space (X, τ), the cardinality of LCT(X, τ) is at least \( \mathfrak{c} \).