Diffeologies on Locally Convex Spaces and Smooth Multiplication of Distributions

Abstract

We investigate the canonical and $c^\infty$-diffeologies on Hausdorff locally convex spaces and their applications to Schwartz distributions. We prove that a Hausdorff locally convex space, endowed with its canonical diffeology, is convenient if and only if the canonical map to its internal tangent space at each point is a linear isomorphism. This yields a geometric characterization of Mackey completeness. We also compare several natural diffeologies on locally convex spaces and identify conditions under which they are preserved under completion, dualization, and the formation of inductive limits. As an application, we realize the space of microlocally multipliable distributions as a diffeological colimit and show that H\"ormander-admissible multiplication is smooth. This establishes a framework for nonlinear distribution theory beyond the classical manifold setting.