The closed graph theorem is the open mapping theorem

TL;DR

This paper explores a duality-based extension of the classical closed graph and open mapping theorems, revealing a deep connection between these fundamental results in functional analysis.

Contribution

It introduces a duality framework that interchanges the extensions of the closed graph and open mapping theorems, broadening their applicability.

Findings

Established a duality linking the closed graph and open mapping theorems.

Extended classical theorems to new contexts via duality.

Revealed a fundamental symmetry between these theorems.

Abstract

We extend the closed graph theorem and the open mapping theorem to a context in which a natural duality interchanges their extensions.