On the Topology Induced by Mixtures of Exponentials

Abstract

It is shown that if X is a real vector space of uncountable dimension then the coarsest topology on X for which the function $$x \mapsto \smallint _{x*} e^{\xi (x)} d\mu (x)$$ is continuous whenever μ is a measure on the dual space X * integrating the integrands is strictly coarser than the finest locally convex topology. This is derived from an inequality relating the averages of such a 'mixture of exponentials' on the vertices and facet midpoints, respectively, of a 'generalized octahedron' in a finite-dimensional space (Lemma 1).