Completeness of Holder Space How do I show that the space of complex-valued functions on $[0,1]$ such that
$$|f(x) -f(y)| < C|x-y|^\frac{1}{2}$$ with norm
$$\|f\| = \sup_{x \in [0,1]} |f(x)| + \inf C$$ where inf is over the constants such that the holder bound holds, is a Banach space? $C$ depends on the function. 
 A: Note that $\inf C$ for a function $f$ is better described as $$\sup_{x, y \in [0,1], x \ne y} \frac{|f(x) - f(y)|}{|x - y|^{1/2}}$$
Also notice that $f$ is Hölder continuous if and only if this supremum is finite.

Let $(f_n)_n$ be a Cauchy sequence of Hölder continuous functions w.r.t $\|\cdot\|$. 
Since $\|\cdot\|_\infty \le \|\cdot\|$ it follows that $(f_n)_n$ is Cauchy w.r.t $\|\cdot\|_\infty$. Recall that $(C[0,1], \|\cdot\|_\infty)$ is a Banach space so there exists $f \in C[0,1]$ such that $f_n \xrightarrow{n\to\infty} f$ uniformly (namely, $f_n$ is precisely the pointwise limit of $f_n$).
We need to show that $f_n \xrightarrow{\|\cdot\|} f$ and that $f$ is  Hölder continuous.
Let $\varepsilon > 0$. $(f_n)_n$ is Cauchy w.r.t $\|\cdot\|$  so there exists $N \in \mathbb{N}$ such that $m, n \ge N \implies \|f_m - f_n\| < \frac\varepsilon3$.
In particular we have $\|f_m - f_n\|_\infty < \frac\varepsilon3$ so letting $m\to\infty$ we get $\|f - f_n\|_\infty \le \frac\varepsilon3$.
On the other hand, for all $x,y \in [0,1], x \ne y$ we also have
$$\frac{|(f_m(x) - f_n(x)) - (f_m(y) - f_n(y))|}{|x - y|^{1/2}} < \frac\varepsilon3$$
Letting $m\to\infty$ we get
$$\sup_{x, y \in [0,1], x \ne y}\frac{|(f(x) - f_n(x)) - (f(y) - f_n(y))|}{|x - y|^{1/2}} \le \frac\varepsilon3$$
Putting this together, we get that for all $n \ge N$ holds $\|f - f_n\| \le \frac{2\varepsilon}3 < \varepsilon$. It follows that $f_n \xrightarrow{\|\cdot\|} f$.
Now, for any $n \ge N$ we have
$$\|f\| \le \|f - f_n\| + \|f_n\| < \varepsilon + \|f_n\| < +\infty$$
so we conclude that $f$ is Hölder continuous.
Therefore, the space of Hölder continuous functions equipped with $\|\cdot\|$ is a Banach space.
Also see here for a similar proof concerning a slightly different norm on Hölder continuous functions.
