Duality of Discrete Topological Vector Spaces

TL;DR

This paper explores the structure and categorical equivalences of discrete topological vector spaces over a field, showing their isomorphisms to certain ordinal-indexed spaces and establishing category equivalences.

Contribution

It characterizes discrete topological vector spaces over a field as ordinal-indexed spaces and proves the categorical equivalence between different classes of these spaces.

Findings

Discrete topological vector spaces are isomorphic to ^{eta} for some ordinal .

Under certain conditions, these spaces are isomorphic to ^{(eta)}.

Categories of these vector spaces are equivalent.

Abstract

For a field $\ef$, the discrete topological vector spaces over $\ \ef$ are essentially of the form $\ef^{\alpha}$ where $\alpha$ is an ordinal. With additional appropriate properties, they are isomorphic to $\ef^{(\beta)}$ where $\beta$ is again an ordinal. Finally, the categories of the vector spaces of the the first and the second type are equivalent.