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$H$ 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?

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

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  • $\begingroup$ Just to be clear, we define our functions $u:[0,T]\to H$ such that $u$ is continuous? $\endgroup$
    – Ian Coley
    Apr 16, 2013 at 17:06
  • $\begingroup$ @FrankMcGovern I think $\lVert u(t) - u(s) \rVert_H \to 0$ as $t \to s$. $\endgroup$
    – markus
    Apr 16, 2013 at 17:08

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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.

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