Arzelà–Ascoli Theorem

For the rest of this section, let $K=(K,d_K)$ be a compact metric space and $(X,d_X)$ be a metric space, then we will consider the space of continuous functions

Arzelà-Ascoli Theorem

Let $K$ be a compact metric space, and $X$ be a complete metric space. A subset $\mathcal{H}$ of ${\mathcal{C}}^0(K,X)$ is relatively compact on ${\mathcal{C}}^0(K,X)$ iff $\mathcal{H}$ is equicontinuous and the sets

are relatively compact in $X$ for all $z\in K$

Let $X$ and $K$ be compact metric spaces, then a subet $\mathcal{H}$ of ${\mathcal{C}}^0(K,X)$ is relatively compact iff $\mathcal{H}$ is equicontinuous.

Let $K$ be compact metric space, and $X$ be a complete metric space, the sequence $(f_k)$ in the space ${\mathcal{C}}^0(K,X)$ congerves pointwise to the function $f:K\to X$. If $\mathcal{H}:=\{f_k\mid k\in \mathbb{N}\}$ is equicontinuous then $f$ is continuous and $f_k$ converges uniformly to $f$ or $f_k$ converges in ${\mathcal{C}}^0(K,X)$

Arzelà-Ascoli Theorem to ${\mathbb{R}}^n$

Let $K$ be a compact metric space. A subset $\mathcal{H}$ of ${\mathcal{C}}^0(K,{\mathbb{R}}^n)$ is relatively compact on ${\mathcal{C}}^0(K,{\mathbb{R}}^n)$ iff $\mathcal{H}$ is equicontinuous bounded in ${\mathcal{C}}^0(K,{\mathbb{R}}^n)$

Arzelà–Ascoli Theorem from ${\mathbb{R}}^m$ to ${\mathbb{R}}^n$

Let $A\subseteq {\mathbb{R}}^m$ be compact and let $B\subseteq \mathcal{C}(A,{\mathbb{R}}^n)$. If $B$ is bounded and equicontinuous, then any sequence in $B$ has a uniformly convergent subsequence.

Thus we have a characterization of sequential compactness in $\mathcal{C}(A,{\mathbb{R}}^n)$, when $A$ is compact.