# An improvement of the Kolmogorov-Riesz compactness theorem

## TL;DR

This paper presents a concise proof that enhances the Kolmogorov-Riesz theorem, which characterizes compact subsets in Lebesgue spaces, thereby advancing the theoretical understanding of functional analysis.

## Contribution

It provides a new, shorter proof of Sudakov's result, significantly improving the classical Kolmogorov-Riesz theorem.

## Key findings

- Simplified proof of Sudakov's result
- Improved characterization of compact sets in Lebesgue spaces
- Enhanced theoretical framework for functional analysis

## Abstract

The purpose of this short note is to provide a new and very short proof of a result by Sudakov, offering an important improvement of the classical result by Kolmogorov-Riesz on compact subsets of Lebesgue spaces.