i know that $ L^2 (0, T; H^1_0(\Omega))$ and $C (0, T; H^1_0(\Omega))$ when T is finite are banach spaces can we say the same thing if $T =\infty$? i read also in evans book that $ L^{2}(0, T; H_{0}^{1}(\Omega))$ is equipped with norm:$$ \int_0^\infty \left\Vert u(t) \right\Vert ^2 _{H^1_0(\Omega)} dt$$
so can i write that $ L^2 (0, T; H^1_0(\Omega))$ is equipped with norm :$$ \int_0^\infty \left\Vert u(t) \right\Vert ^2 _{H^1_0(\Omega)} dt$$ and i find in Evans book definition of the space $C^0([0, T] ; V) $ which is equipped with norm :
$$\|f\|_{C^0([0, T] ; V)}=\max _{t \in[0, T]}\|f(t)\|_V . $$
but for the space $C^{0}\left([0, \infty[ ; V\right)$ what norm i should consider for this space? Thanks