An improvement of the Kolmogorov-Riesz compactness theorem (Q2003392)

The well-known Kolmogorov-Riesz compactness criterion establishes a necessary and sufficient condition under which a set \(M\subset L^p(\mathbb{R})\) \((1\le p< \infty)\) is totally bounded. Subsequently it was shown by \textit{V. N. Sudakov} [Usp. Mat. Nauk 12, No. 3(75), 221--224 (1957; Zbl 0079.12805)] that the boundedness requirement for \(M\) in the sufficiency part is superfluous. In this paper, the authors give an elementary proof of this result.