Arzela-Ascoll Theorem

Theorem (Arzela-Ascolli)

Let \(A \subset M\) be compact, then \(\mathcal{B} \subset \mathcal{C}_b\) is compact if and only if it is closed, pointwise compact, and equicontinuous.

Proof

(\(\impliedby\)) By the Volzano-Weierstrass theorem, it suffices to show that \(\mathcal{B}\) is sequentially compact.

• Claim: For every sequence of functions \(f_n \in \mathcal{B}\), there exists convergent subsequence of \(f_n\).

Since \(A\) is compact, it is totally bounded, thus for all \(\delta >0\), there is a finite set

\[ C_\delta = \{ y_{\delta_1}, y_{\delta_2}, \cdots, y_{\delta_n}\} \tag{1}\]

such that \(\bigcup_{i=1}^n B_{y_{\delta_i}, \delta} \supset A\). Let

\[ C = \bigcup_{i=1}^n C_{\frac{1}{n}} \]

and since \(C\) is countable, we relabel it as \(C = \{x_1, x_2, \cdots\}\). For a sequence of functions \(f_n \in \mathcal{B}\), since \(\mathcal{B}\) is pointwisely compact, we can construct a subsequence \(f_{1, j}\) of \(f\) such that \(f_{1, j}(x_1)\) converges. Inductively, for every \(k \in \mathbb{N}\) we construct a subsequence \(f_{k+1, j}\) of \(f_{k, j}\) such that \(f_{k+1, j}(x_{k+1})\) converges.

Define \[ g_n = f_{n, n} \] , then we observe \(g_n (x_i)\) converges for all \(i \in \mathbb{N}\).

It is enough to finish our proof by showing that \(g\) is uniformly convergent, i.e.,

• Claim: For each \(x \in A\), \(\forall \varepsilon >0\), \(\exists n_0 \in \mathbb{N}\) such that \(m, n \geq n_0\) implies \(\rho(g_n(x), g_m(x)) < \varepsilon\).

Since \(g_n\) is equicontinuous, we could choose \(\delta > 0\) such that for every \(i\), \(d(x, y) < \delta\) implies \[ \rho(g_i (x), g_i(y)) < \frac{\varepsilon}{3}. \]

Depending to our choice of \(\delta\), we construct a finite set \(C_\delta\) as Equation 1. For a given \(x\), take \(y \in C_\delta\) such that \(d(x, y) < \delta\). Since \(g_i\) is convergent pointwisely, take \(n_0 \in \mathbb{N}\) such that \(n, m \geq n_0\) implies \[\rho(g_n (y), g_m(y)) < \frac{\varepsilon}{3}.\] Then, by the triangle inequality, we obtain

\[\rho(g_n(x), g_m(x)) \leq \rho(g_n(x), g_n(y)) + \rho(g_n(y), g_m(y)) + \rho(g_m(y), g_m(x))<\varepsilon.\]