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

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