Topology on diffeological vector spaces

TL;DR

This paper investigates the conditions under which the D-topology renders diffeological vector spaces into topological vector spaces, identifying cases where this holds and proposing the study of almost topological vector spaces.

Contribution

It demonstrates that the D-topology makes many diffeological vector spaces into topological vector spaces using $k_$-space theory, but not universally, and introduces the concept of almost topological vector spaces.

Findings

D-topology induces topological vector space structure for a large class of diffeological vector spaces.

Counterexamples show that this is not true for all diffeological vector spaces.

Proposes the study of almost topological vector spaces as a new class.

Abstract

It is expected that the $D$-topology makes every diffeological vector space into a topological vector space. We show that it is the case for a large class of diffeological vector spaces via $k_\omega$-space theory, but not so in general. The paper also proposes the study of a class of almost topological vector spaces.