The Kolmogorov-Riesz compactness theorem

TL;DR

This paper demonstrates that the Arzela-Ascoli and Kolmogorov compactness theorems are derived from a fundamental lemma on metric space compactness, exploring their historical context and relation to Helly's theorem.

Contribution

It reveals the underlying connection between key compactness theorems in analysis through a simple lemma, providing new insights into their foundational relationships.

Findings

Unified proof of Arzela-Ascoli and Kolmogorov theorems

Discussion of the relation to Helly's theorem

Historical analysis of the Kolmogorov compactness theorem

Abstract

We show that the Arzela-Ascoli theorem and Kolmogorov compactness theorem both are consequences of a simple lemma on compactness in metric spaces. Their relation to Helly's theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov compactness theorem.