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

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