On the Closed Graph Theorem and the Open Mapping Theorem

TL;DR

This paper explores the relationship between the Closed Graph Theorem and the Open Mapping Theorem in topological algebra, clarifying their conditions, equivalences, and exceptions in various topological spaces.

Contribution

It provides a detailed analysis of the conditions under which the Closed Graph Theorem and the Open Mapping Theorem are equivalent or hold separately in topological groups, TVS's, and LCS's.

Findings

Theorems are equivalent under certain conditions.

Exceptions to the theorems are identified and explained.

Key versions of the theorems are stated with references.

Abstract

Let $E,F$ be two topological spaces and $u:E\rightarrow F$ be a map. \ If $F$ is Haudorff and $u$ is continuous, then its graph is closed. \ \ The Closed Graph Theorem establishes the converse when $E$ and $F$ are suitable objects of topological algebra, and more specifically topological groups, topological vector spaces (TVS's) or locally vector spaces (LCS's) of a special type. The Open Mapping Theorem, also called the Banach-Schauder theorem, states that under suitable conditions on $E$ and $F,$ if $ v:F\rightarrow E$ is a continuous linear surjective map, it is open. \ When the Open Mapping Theorem holds true for $v,$ so does the Closed Graph Theorem for $u.$ \ The converse is also valid in most cases, but there are exceptions. \ This point is clarified. Some of the most important versions of the Closed Graph Theorem and of the Open Mapping Theorem are stated without proof but with the detailed reference.