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](https://core.ac.uk/download/pdf/82559503.pdf) 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.



Who are the owners of the compactness theorem in $L^p(\Bbb R^d)$?
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](https://zbmath.org/?q=an:0084.10402).


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




---


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


1908


*Fréchet, M.*, [**Sur quelques points du calcul fonctionnel.**](http://dx.doi.org/10.1007/BF03018603), Palermo Rend. 22, 1-74 (1906); auch sep. (Thése) Paris: Gauthier-Villars (1906). [ZBL37.0348.02](https://zbmath.org/?q=an:37.0348.02).


1937


*Fréchet, M.*, [Sur les ensembles compacts de fonctions de carrés sommables](http://pub.acta.hu/acta/showCustomerArticle.action?id=5577&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=5cd0a030576a3c76&style=), Acta Litt. Sci. Univ., Szeged, Sect. Sci. math. 8, 116-126 (1937). [ZBL63.0200.02](https://zbmath.org/?q=an:63.0200.02).


