The classical Ascoli-Arzelà theorem could be stated as follows:
Let $K$ be a compact metric space and let $\mathcal{H}$ be a bounded subset of $C(K)$ - the space of continuous functions over $K$ with values in $\mathbb{R}$ or $\mathbb{C}$. Assume that $$ \forall \epsilon > 0 \text{ } \exists \delta > 0: d(x, y) < \delta \implies |f(x) - f(y)| < \epsilon \text{ } \forall f \in \mathcal{H}.$$ Then $\mathcal{H}$ is a relatively compact subset of $C(K)$.
Question: I want to restate this theorem using certain approximation quantities and I wanted to be sure that what I have stated is correct. Before I do, I will give precise definitions of the approximation quantities I will use:
Modulus of continuity: Let $f(x)$ be defined on a compact set $K$. The function $$\omega(f, \delta) = \sup_{x, y \in K: d(x, y) \leq \delta} |f(x) - f(y)|$$ is called the modulus of continuity of $f$ on $K$. --- the notion of equicontinuity should be connected to the modulus of continuity being zero for some $\delta_{f}$.
nth entropy numbers: Let $M$ be a bounded subset of $X$. Then for $n \in \mathbb{N}$ define the nth entropy number of $M$ as follows:
$$\epsilon_{n}(M) = \inf \{ \epsilon > 0: \text{ there exists } q \leq n \text{ points } x_{1}, x_{2}, \cdots, x_{q} \text { in } X \text{ s.t } M \subset \bigcup_{i = 1}^{q} B(x_{i}, \epsilon) \}$$
- A bounded subset $M \subset X$ is relatively compact if and only if $$\lim_{n \to \infty} \epsilon_{n}(M) = 0$$
Now, the Ascoli-Arzelà may be stated as follows:
Ascoli-Arzelà:Let $K$ be a compact metric space and let $\mathcal{H}$ be a bounded subset of $C(K)$. Then $$\omega(f, \delta) = 0 \text{ } \forall \text{ } f \in \mathcal{H} \implies \lim_{n \to \infty} \epsilon_{n} (\mathcal{H}) = 0$$