Arzela-Ascoli and compactness in $C(X), l^p, L^p$

Question

Arzela-Ascoli and compactness in $C(X), l^p, L^p$

• $C(X)$ with the uniform norm and $X$ is a compact metric space, a closed and bounded set in $C(X)$ is compact if and only if it is equicontinuous.

• a closed and bounded subset in $l^p$ $(1\leq p<\infty)$is compact if and only if it is equisummable, that is for each $\epsilon > 0$ there exists a $N$ such that $\sum_{n=N}^\infty |x_n|^p \leq \epsilon$ for each $x = \{x_n\}$ in this set.

Questions:

1. What is the connection between equicontinuous and equisummable, and how to connect between $C(X)$ with $l^p$, is there something similar for $L^p$?

2. What is the intuition behind equicontinuous and equisummable? From my own understanding, in a infinite dimensional space say $l^p$ $(p<\infty)$, because the metric or norm has to measure the quantity in every single one of the dimensions, there can be too much variation in each dimension for a closed and bounded set to be compact in the metric topology. The concept of equisummable would make this variation small. Is this correct? And what about equicontinuous?

Thank you very much!

Answer 1

Question 1: I don't know of any connection between equicontinuity and equisummability. There is indeed something similar for $L^p$, sometimes known as the Kolmogorov–Riesz theorem:

A subset $\mathcal{F}$ of $L^p(\mathbb{R}^n)$ (with $1\le p<\infty$) is compact if and only if it is closed, bounded, equi-integrable in the sense that for every $\varepsilon>0$ there is some $R$ so that for every $f\in\mathcal{F}$, $$\int_{|x|>R}|f(x)|^p\,dx<\varepsilon,$$ and one more condition is satisfied, a sort of equicontinuity in the mean: for every $\varepsilon>0$ there is some $\delta>0$ so that, for every $f\in\mathcal{F}$ and every $y\in\mathbb{R}^n$ with $|y|<\delta$, $$\int_{\mathbb{R}^n}|f(x+y)-f(x)|^p\,dx<\varepsilon.$$

Question 2: The intuition for equisummability is most easily explained using a non-equisummable sequence: Let $e_n$ be the sequence with all zeros, except for $1$ in the $n$th position. It has no convergent subsequence in $\ell^p$, and the reason for that is that there is “mass escaping to infinity”. The equisummability condition stops that from happening.

As for equicontinuity, without it you can have examples like $\arctan(nx)$ which converges (as $n\to\infty$) to a discontinuous function. Here, the problem is that the continuity of each function is progressively worse. Equicontinuity stops that from happening.

Shameless plug: See a paper on the Kolmogorov–Riesz theorem that I cowrote.