Localized locally convex topologies

TL;DR

This paper introduces and analyzes localized convex topologies on vector spaces, motivated by ill-posed PDEs, revealing their functional properties and implications for divergence equations and distributional solutions.

Contribution

It develops a framework for localized topologies, studies their properties, and provides an abstract existence theorem for divergence equations under various conditions.

Findings

Localized topologies are sequential but not Fréchet-Urysohn, barrelled, or bornological.

The topologies are semireflexive iff members of the convex family are compact.

An abstract existence theorem characterizes divergence solutions for different regularities and boundary conditions.

Abstract

Motivated by ill-posed PDEs such as $\mathrm{div} (v) = F$ we study locally convex topologies $\mathcal{T}_{\mathcal{C}}$ on real vector spaces $X$ that are a ``localized'' version of a locally convex topology $\mathcal{T}$ to members of a family $\mathcal{C}$ of convex subsets of $X$. The distributions $F$ arising as $\mathrm{div} (v)$ are expected to be the members of the dual of well-chosen $X$ with respect to an appropriate localized topology $\mathcal{T}_{\mathcal{C}}$. In this work, the emphasis is on studying the functional analytic properties of $\mathcal{T}_{\mathcal{C}}$, according to those of $\mathcal{T}$ and $\mathcal{C}$. For instance, we show that in all foreseen applications, $\mathcal{T}_{\mathcal{C}}$ is sequential but none of Fr\'echet-Urysohn, barrelled, and bornological. These awkward phenomena are illustrated explicitly on a specific example corresponding to the distributional divergence of continuous vector fields in $\mathbb{R}^m$. We also show that, essentially, $\mathcal{T}_{\mathcal{C}}$ is semireflexive if and only if members of $\mathcal{C}$ are $\mathcal{T}$-compact. This leads to an abstract existence theorem, thereby establishing a general scheme for characterizing those $F$ such that $\mathrm{div} (v) = F$ for various classes of regularity of $v$, various classes of domains, and various boundary conditions.