A simple proof of the closed graph theorem

TL;DR

This paper presents a concise proof of the closed graph theorem for Hilbert spaces using the uniform boundedness principle, with an extension to Banach spaces, simplifying the classical understanding of operator boundedness.

Contribution

It introduces a new, simplified proof of the closed graph theorem leveraging the uniform boundedness principle, applicable to both Hilbert and Banach spaces.

Findings

The proof confirms that closed linear operators on Hilbert spaces are bounded.

The approach simplifies the classical proof of the closed graph theorem.

Extension of the proof to Banach spaces broadens its applicability.

Abstract

Assume that $A$ is a closed linear operator defined on all of a Hilbert space $H$. Then $A$ is bounded. A new short proof of this classical theorem is given on the basis of the uniform boundedness principle. The proof can be easily extended to Banach spaces.