Hahn--Banach Theorem and Duality Theory on non-Archimedean Locally Convex Spaces

TL;DR

This paper extends the Hahn--Banach theorem and duality theory to non-Archimedean locally convex spaces over local fields, broadening the theoretical framework for such spaces.

Contribution

It introduces extensions of Hahn--Banach and Iwasawa-type duality theorems to various classes of non-Archimedean locally convex spaces over local fields, valuation rings, and residue fields.

Findings

Extended Hahn--Banach theorem to seminormed non-Archimedean spaces

Established duality analogues for multiple classes of locally convex spaces

Provided theoretical foundations for non-Archimedean functional analysis

Abstract

Let $k$ be a local field with valuation ring $O_k$ and residue field $\overline{k}$. We extend Hahn--Banach theorem for the class of seminormed $k$-vector spaces to several classes of locally convex spaces and subspaces over $k$, $O_k$, and $\overline{k}$. We establish analogues of Iwasawa-type duality for several classes of locally convex spaces over $k$, $O_k$, and $\overline{k}$.