Applications Arzela-Ascoli theorem

Question

Dears,

I am looking for some "nice" applications of the Arzelá-Ascoli theorem for the Banach space of continuous functions $x:[a,b]\longrightarrow X$, where $X$ is a Banach space.

I known this result have important applications in the analyse of certain integral equations. Somebody know anothers applications?

Thanks.

Answer 1

• The identity mapping $\iota: C^m([0,1]) \hookrightarrow C^{m-1}([0,1])$ is a compact operator. Here the norm is $\|f\|_{C^k} =\sum_{i=0}^k \max_{[0,1]}|f^{(i)}(x)|$.
• $\mathcal{F} = \{u \in C^2(U) \ | \ \Delta u = 0 \text{ in } U, \|u\|_{L^\infty}\le M\}$ (for $K$ compact subset of $U$) is pre-compact in $C(K)$.
• Define $\tau_h(f)(x) = f(x+h)$. If $f\in L^\infty(\mathbb{R})$ and $\|\tau_h(f) - f\|_{L^\infty} \to 0$, then $f$ is equal to a uniformly continuous function almost everywhere.
• The "Arzela-Acoli" for $L^p$ spaces involves an application of the normal Arzela-Ascoli.