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
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.