Problem 2

Exercise 1.2 (Hölder condition for compactness)   Show that the following set is compact

$$A = \{ f \in C(X) \;\vert\; \Vert f\Vert \leq 1, H_\alpha(f) \leq 1 \}$$ (1.5)

where $(X,\rho)$ is a compact metric space and

$$H_\alpha(f) := \sup_{x\neq y}\frac{\vert f(x)-f(y)\vert}{\rho(x,y)^\alpha}$$ (1.6)

Proof. To apply Arzelà-Ascoli, lets show $A$ is closed, bounded, and equicontinuous. The first two are obvious, so we focus on the last. Let $\epsilon>0$ be given. If $\rho(x,y)<\epsilon^{1/\alpha}$, then $\rho^\alpha(x,y) \leq \epsilon$ and

$$H_\alpha(f)\leq 1 \implies \frac{\vert f(x)-f(y)\vert}{\rho^\alph... ...s \vert f(x)-f(y)\vert\leq \rho^{\alpha}(x,y) \leq \epsilon \quad\forall f\in A$$ (1.7)

Therefore, $\delta=\epsilon^{1/\alpha}$ is an appropriate equicontinuity constant. $\qedsymbol$