$H^\infty$ with the norm $\| \cdot \|_\infty$ is a (complex) Banach space

Question

Where can I find the proof of this lemma?

$H^\infty$ with the norm $\| \cdot \|_\infty$ is a (complex) Banach space

Where $H^\infty = \{f \in H(U) \mid \|f\|_\infty \lt \infty\}$ with $U$ be the open unit disk in the complex plane, and $H(U)$ is the set of holomorphic functions in $U$.

Answer 1

You know the proof that $C(X)$ is a Banach space, right? The completeness comes from the fact that a uniform limit of continuous functions is continuous. So do the same thing, except using the fact that a uniform limit of holomorphic functions is holomorphic.