We show that the Arzelà-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. IntroductionCompactness results in the spaces L p (R d ) (1 ≤ p < ∞) are often vital in existence proofs for nonlinear partial differential equations. A necessary and sufficient condition for a subset of L p (R d ) to be compact is given in what is often called the Kolmogorov compactness theorem, or Fréchet-Kolmogorov compactness theorem. Proofs of this theorem are frequently based on the Arzelà-Ascoli theorem. We here show how one can deduce both the Kolmogorov compactness theorem and the Arzelà-Ascoli theorem from one common lemma on compactness in metric spaces, which again is based on the fact that a metric space is compact if and only if it is complete and totally bounded.Furthermore, we trace out the historical roots of Kolmogorov's compactness theorem, which originated in Kolmogorov's classical paper [18] from 1931. However, there were several other approaches to the issue of describing compact subsets of L p (R d ) prior to and after Kolmogorov, and several of these are described in Section 4. Furthermore, extensions to other spaces, say L p (R d ) (0 ≤ p < 1), Orlicz spaces, or compact groups, are described. Helly's theorem is often used as a replacement for Kolmogorov's compactness theorem, in particular in the context of nonlinear hyperbolic conservation laws, in spite of being more specialized (e.g., in the sense that its classical version requires one spatial dimension). For instance, Helly's theorem is an essential ingredient in Glimm's ground breaking existence proof for nonlinear hyperbolic systems [14]. We show below that Helly's theorem is an easy consequence of Kolmogorov's compactness theorem. Preliminary resultsAn ε-cover of a metric space is a cover of the space consisting of sets of diameter at most ε. A metric space is called totally bounded if it admits a finite ε-cover for every ε > 0. It is well known that a metric space is compact if and only if it is complete and totally bounded (see, e.g., [34, p. 13]). Since we are interested in compactness results for subsets of Banach spaces, we may, and shall, concentrate our attention on total boundedness.

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Norm closed (or weakly closed) Jordan algebras of self-adjoint operators on a Hilbert space were initially studied by Topping, Effros, and Stormer [15], [4], [12], [13]. These works are very “spatial”, in that the algebras are considered in one given representation. The introduction of their abstract counterparts, the JB- and JBW-algebras, has led to an increased interest in this subject. The author hopes this paper will support the view that a more “space-free” approach is fruitful, even if only the “concrete” algebras are under study. In accordance with this view, a “JC-algebra” in this paper will mean a normed Jordan algebra over the reals, which is isometrically isomorphic to a norm closed Jordan algebra of self-adjoint operators.Some of the results in this paper are closely related to, or rewordings of, results in the above-mentioned papers. However, I feel that the present approach is sufficiently different to be of interest in itself. In particular, many of the technical difficulties associated with earlier approaches are avoided.

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