There are two points which are not clear for me in Brezis's book Functional Analysis: The space $L^2 (0, \infty; H^1_0(\Omega))$, what is its definition?
I found in Evans book the definition of space $L^2 (0, T; H^1_0(\Omega))$ : it is the space of measurable functions u such that $$ \int_0^\infty \left\Vert u(t) \right\Vert ^2 _{H^1_0(\Omega)} ~\mathrm{d}t < \infty.$$ Can I just put $T=\infty$ to get defintion of $L^2 (0, \infty; H^1_0(\Omega))$?
What about this space: $C((0,\infty,L^{p}(\Omega))$? How can one define it? For $T$ finite and for $C((0,T,L^{p}(\Omega))$ it is space of strongly continious function such that: $$\|u(t)-u(s)\|_{L^{p}}\to 0 \ as t \to s,$$ but I dont think it will be true for $T=\infty$.
My second question for $T$ finite we have that $C((0,T,L^{p}(\Omega))$ is a Banach space, but For $T =\infty$ this is false? Please if you have any references where I can find the definition of these spaces it will be great help. Thank you.