Who are the owners of the compactness theorem in $L^p(\Bbb R^d)$?

Question

As the title says, I am interested to know Who are the owners of the compactness theorem in $L^p(\Bbb R^d)$. There is some confusion in the literature.

Let recall that the compactness theorem in $L^p(\Bbb R^d)$ is somewhat a generalization of the Ascoli-Arzelà compactness theorem for space $C(X)$ of continuous functions on a compact metric space $(X,d)$.

The theorem can be phrased as follows for $1\leq p<\infty$

Theorem A set $\mathcal{F}\subset L^p(\Bbb R^d)$ is compact if and only if

1. (Boundedness) $\sup_{u\in \mathcal{F}}\|u\|_{L^p(\Bbb R^d}<\infty,$
2. ($p$-equicontinuity) $$\lim_{|h|\to 0} \sup_{u\in \mathcal{F}}\|u(\cdot+h)-u(\cdot)\|_{L^p(\Bbb R^d}=0,$$
3. $p$-tighness $$\lim_{R\to \infty} \sup_{u\in \mathcal{F}}\int_{|x|>R}|u(x)|^pd x=0,$$

The book by Haim Brezis (Functional Analysis, Sobolev Spaces, and Partial Differential Equations) names this Theorem as "the Riesz-Fréchret-Kolmogorov Theorem"

However, further research led me to this article with more historical details and the name of Fréchet does not appear therein. The theorem is thus merely named as "Kolmogorov-Riesz Theorem".

Is there any reason why Brezis added the name of Fréchet?

Does any experience professor here have some additional information about this? I am actually writing a manuscript and I would like to make sure the right owners are clearly mentioned.

Answer 1

I consulted the compendum on such topics: Dunford & Schwartz.

Dunford, Nelson; Schwartz, Jacob T., Linear operators. I. General theory. (With the assistence of William G. Bade and Robert G. Bartle), Pure and Applied Mathematics. Vol. 7. New York and London: Interscience Publishers. xiv, 858 p. (1958). ZBL0084.10402.

The compactness results for $L_p$ are IV.8.18, IV.8.20, IV.8.21. From the notes at the end of chapter IV, it seems:

Fréchet [1908] characterized compact sets in $L_2[0,1]$

Kolmogorov [1931] characterized compact sets in $L_p(S)$, $1 < p < \infty$, $S$ a bounded set in finite-dimensional Euclidean space

Tamarkin [1932] extended to unbounded sets

(There were also papers with cases $p=\infty, p=1, 0<p<1$.)

M. Riesz [1933] gave simpler proofs. (Dunford & Schwatz use the proof of Riesz in their text.)

Takahashi [1934] did Orlicz space $L_\Phi$

Fréchet [1937] gave different compactness criteria for $L_p$

They note a further generalization to $L_p(G)$ where $G$ is a locally compact group

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So, to answer the main question: Fréchet has two contributions

1908

Fréchet, M., Sur quelques points du calcul fonctionnel., Palermo Rend. 22, 1-74 (1906); auch sep. (Thése) Paris: Gauthier-Villars (1906). ZBL37.0348.02.

1937

Fréchet, M., Sur les ensembles compacts de fonctions de carrés sommables, Acta Litt. Sci. Univ., Szeged, Sect. Sci. math. 8, 116-126 (1937). ZBL63.0200.02.