A new and simple proof of Schauder's theorem

TL;DR

This paper presents a new, concise, and elementary proof of Schauder's theorem, establishing the equivalence of compactness for a bounded linear operator and its adjoint in Banach spaces without complex tools.

Contribution

The paper introduces a simplified proof of Schauder's theorem that relies solely on basic functional analysis principles, avoiding advanced theorems like Arzela--Ascoli.

Findings

Proof is shorter and more elementary than existing proofs

Avoids reliance on the Arzela--Ascoli theorem

Provides a clearer understanding of the theorem's core concepts

Abstract

Schauder's theorem asserts that a bounded linear operator between Banach spaces is compact if ad only if its adjoint is. We give a new proof of this result, which is both short and completely elementary in the sense that it does not depend on anything beyond basic functional analysis, i.e., the Hahn--Banach theorem and some of its consequences; in particular, we avoid the Arzela--Ascoli theorem (and any kind of related diagonal argument).