Lipschitz vector spaces

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 $E$.