arXiv subjects

Tullio Valent

Publications and source records attributed to Tullio Valent.

3 recordsLinked to original sources

arXiv2024

Lipschitz vector spaces

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 $E$.

math.GN↗

arXiv2024

Weak Lipschitz structures and their connections with the topological structures

Two approaches to Lipschitz structures for any set are presented, studied and compared. The first approach is similar to the one proposed in Fraser, Jr. R. B., Axiom systems for Lipschitz structures, Fundamenta Mathematicae, (1970), where Lipschitz structures are defined as families of pseudo-metrics satisfying suitable conditions. The other one, here introduced, is expressed by using weak pseudo-metrics, which (unlike the pseudo-metrics) do not necessarily vanish on the whole of the diagonal of the cartesian product of the considered set. In this case we will talk about weak Lipschitz structures. Since all topological structures are defined by a a family of weak pseudo-metrics (as we will show in Section 4) we can find some connections between topological structures and weak Lipschitz structures, and a link between continuous maps and weak Lipschitz maps. A central part of this paper is devoted to the weak Lipschitz uniformity defined by a weak Lipschitz structure, which is introduced in Section 8. A notion of uniform continuity with respect to weak Lipschitz uniformities is proposed and studied. In particular, we prove that the weak Lipschitz maps acting between two weak Lipschitz spaces are uniformly continuous with respect to the weak Lipschitz uniformities defined by the respective weak Lipschitz structures.

math.GN↗

arXiv2024

Real structures. An introduction to a general approach

In this paper we attempt to present a very general approach to the study of structures (somehow) defined on a set $X$ by a family of maps $d: X \times X \mapsto \mathbb{R}^+$. It will be shown how the assignment of a preorder $\prec_{\Pi}$ on a set $\Pi$ of families of maps from $X \times X$ into $\mathbb{R}^+$ defines a structure on $X$. The structures obtained in this way will be called \emph{real structures}. For real structures on two different sets we study when they are \emph{of the same type}. An answer to this question will allow to introduce the notion of \emph{morphism}, and then to give the definition of the initial real structure with respect to a family of maps, and so also the definition of product real structure. Various examples of preorders, and hence of real structures, will be exhibited and discussed. A few examples of morphisms will be proposed.

math.GM↗