Let $(C[a, b], \|\cdot\|_\infty)$ be the usual Banach space of continuous functions on $[a, b]$ and for $\alpha\in(0,1]$ and $f\in C[a, b]$ define
$$
[f]_\alpha = \sup_{x,y\in[a,b];x\neq y}\frac{|f(x)-f(y)|}{|x – y|^\alpha}
$$
Let $C^\alpha[a, b]$ be the set of functions $f$ in $C[a, b]$ for which $[f]_\alpha < \infty$, and endow $C^\alpha[a, b]$ with the norm
$$
\|f\|_\alpha = \|f\|_\infty + [f]_\alpha
$$
It is known that $(C^\alpha[a, b], \|\cdot\|_\alpha)$ is a Banach space.
I've been asked to show that the unit ball $B^\alpha := \{f\in C^\alpha[a, b]\ :\ \|f\|_\alpha\le 1\}$ is compact in $(C[a, b], \|\cdot\|)$. Not just precompact, but compact. I've already shown that it's precompact using Arzela-Ascoli, so all that's left is to show that $B^\alpha$ is closed in $(C[a, b], \|\cdot\|_\infty)$.
Suppose $f_n\in B^\alpha$ converges wrt $\|\cdot\|_\infty$ to $f\in C[a, b]$. We know that $\|f_n\|_\alpha \le 1$, and hence $\|f_n\|_\infty \le \|f_n\|_\alpha \le 1$. We can use this to show that $\|f\|_\infty \le 1$ as well. What can we do to show $\|f\|_\alpha\le 1$? This would show that $B^\alpha$ contains its $\|\cdot\|_\infty$-limit points, and hence is closed.
Best Answer
Take a sequence $(f_n)$ in $B^\alpha$ that converges uniformly to some continuous $f$. Take $x,y,z\in [a,b]$ with $x\ne y$. Then $$ |f(z)|\le |f(z)-f_n(z)| + |f_n(z)| $$ and $$ \frac{|f(x)-f(y)|}{|x-y|^\alpha} \le\frac{|f(x)-f_n(x)|}{|x-y|^\alpha} +\frac{|f_n(x)-f_n(y)|}{|x-y|^\alpha} +\frac{|f_n(y)-f(y)|}{|x-y|^\alpha}. $$ Take $\epsilon>0$. Due to uniform convergence, there is $N=N(x,y,z)$ such that $$ |f(z)-f_n(z)| + \frac{|f(x)-f_n(x)|}{|x-y|^\alpha} + \frac{|f_n(y)-f(y)|}{|x-y|^\alpha}\le \epsilon $$ for all $n>N$. This implies for $n>N$ $$ |f(z)| + \frac{|f(x)-f(y)|}{|x-y|^\alpha} \le |f_n(z)| +\frac{|f_n(x)-f_n(y)|}{|x-y|^\alpha} + \epsilon \le \|f_n\|_\alpha + \epsilon\le 1 + \epsilon. $$ Taking the supremum of the left-hand side over $z$, $x\ne y$, yields $$ \|f\|_\alpha \le 1+\epsilon. $$ Now $\epsilon>0$ was arbitrary, hence $\|f\|_\alpha \le 1$.