Lipschitz vector spaces

TL;DR

This paper explores Lipschitz vector structures on topological vector spaces, showing their relation to pseudo-seminorms and establishing that any topological vector structure can be associated with a Lipschitz structure.

Contribution

It introduces the concept of Lipschitz vector structures, proves their relation to existing topologies, and explores their compatibility with locally convex topologies.

Findings

Topological vector spaces are defined by families of pseudo-seminorms.

Any topological vector structure can be associated with a Lipschitz vector structure.

The paper characterizes Lipschitz structures compatible with locally convex topologies.

Abstract

The initial part of this paper is devoted to the notion of pseudo-seminorm on a vector space $E$. We prove that the topology of every topological vector space is defined by a family of pseudo-seminorms (and so, as it is known, it is uniformizable). Then we devote ourselves to the Lipschitz vector structures on $E$, that is those Lipschitz structures on $E$ for which the addition is a Lipschitz map, while the scalar multiplication is a locally Lipschitz map, and we prove that any topological vector structure on $E$ is associated to some Lipschitz vector structure. Afterwards, we attend to the bornological Lipschitz maps. The final part of the article is devoted to the Lipschitz vector structures compatible with locally convex topologies on $E$.