The Kolmogorov-Riesz compactness theorem

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.Comment: This version is lightly revised in response to referee comments. The paper will appear in Expositiones Mathematica