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.

  • 2
    $\begingroup$ Is there any reason not to trust the historical account given in the paper you cite? It's not uncommon for naming conventions to vary by country (and author): another example that comes to mind is the Cauchy-Lipschitz theorem, also known as the Picard-Lindelof theorem. $\endgroup$
    – Leo Moos
    Jun 20 at 10:06
  • $\begingroup$ I never heard the name Fréchet in connection of compactness criteria in spaces of measurable functions. It might be that he has proven some other result using similar ideas for the proof or that he has found the result independently and that Brezis therefore might want to honor him. But if Brezis does not mention this is hard to verify. Besides that, I agree with Leo Moos that the naming conventions in a country often prefer Mathematician of the same country. (For instance, the Banach–Caccioppoli fixed point theorem is outside of Italy usually only attributed to Banach.) $\endgroup$ Jun 20 at 10:15
  • 1
    $\begingroup$ Asserting first the exact statement of a theorem doesn't make you "owner" of a theorem. Names associated to a theorem more or less faithfully reflect its history. $\endgroup$
    – YCor
    Jun 20 at 14:34
  • 1
    $\begingroup$ @YCor, there's a lot of flexibility offered by "more or less", but surely the degree to which names faithfully reflect history is very small! (Else we wouldn't have Stigler's law of eponymy.) $\endgroup$
    – LSpice
    Jun 20 at 14:36
  • 5
    $\begingroup$ "Who are the owners?" is an awkward phrase, and leads to nitpicking. "To whom should I attribute?" might be better focused. $\endgroup$
    – Matt F.
    Jun 20 at 15:00

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

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


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


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.


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