How to show $C(0,T;H)$ is Banach space

Question

$H$ is a Hilbert space. How do I show that the Bochner space $C(0,T;H)$ of continuous $H$-valued functions is a Banach space with the following norm?

$$\lVert u \rVert = \sup_{t \in [0,T]}\lVert u(t) \rVert_{H}$$

Answer 1

You might find it helpful to review the proof that $C(0,T;\mathbb{R})$ is a Banach space. The hardest part is showing the completeness, and it goes in two steps. Suppose that $\{f_n\}$ is a Cauchy sequence; then

1. For each $x$, the sequence $\{f_n(x)\}$ is a Cauchy sequence in $\mathbb{R}$; hence by the completeness of $\mathbb{R}$ it converges to some limit. Call this limit $f(x)$, so that $f_n \to f$ pointwise.

2. Use the fact that $\{f_n\}$ is uniformly Cauchy to show that in fact $f_n \to f$ uniformly.

The proof of the first part goes through with $\mathbb{R}$ replaced by any complete metric space. The second part works with $\mathbb{R}$ replaced by any metric space whatsoever.