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Hi in Evans book I found that The space $C^0\left([0, T] ; V\right)$ is the space of continuous functions from $[0, T]$ with values in $V$ is a Banach space for its natural norm $$ \|f\|_{C^0([0, T] ; V)}=\max _{t \in[0, T]}\|f(t)\|_V . $$

but in brezis book he considers the space $C^{0}\left([0, \infty[ ; V\right)$ and $L^{p}([0, \infty[ ; V)$ can we define these spaces and there norms in the same way if we have $T=\infty$ ? Thanks

RIM
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The space $C^0([0, \infty[; V)$ with the supremum “norm” is not a normed space, not even if $V = \mathbb R$. The problem is that unbounded functions would have an infinitely large “norm”. However, you can consider an analogous norm on the set of continuous and bouded functions from $[0, \infty)$ to $V$.

I am not sure I understand your question regarding $L^P([0, \infty[; V)$ but here the norms are defined similarly as for finite time intervals. You can even start with any measure space. See the Wikipedia page for Bochner spaces, for instance.

Keba
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    Just to clarify, the key point is that $[0,T]$ is compact and $[0,\infty[$ is not. Continuous functions on a compact domain are necessarily bounded and hence have finite norm. – whpowell96 Aug 31 '23 at 15:29
  • @keba ,@whpowell96 ,thank you for your answers, my problem is that i want to show that a function f belongs to $C^0([0, \infty[; V)$ but i dont see how to do that? – RIM Sep 01 '23 at 09:10
  • Well, that depends on the function of course. The best general advice I can give is to follow the definition of (say, sequential) continuity. I guess it make sense to open another question where you describe your concrete example and what you attempted to do. – Keba Sep 01 '23 at 12:32