dual space of $L^2(0,T;H_0^1(\Omega))$

Question

what is the dual space of $L^2(0,T;H_0^1(\Omega))$ with the norm $$\|f\|:= \biggl(\int_0^T \|f\|_{H_0^1(\Omega)}^2\,dt\biggr)^{1/2}.$$

Answer 1

In general, if $X$ is a Banach space such that $X'$ is separable or reflexive and $1/p+1/q=1$, the dual of $L^p(0,T;X)$ is $L^q(0,T;X')$ in the sense that there exists a bijective linear isometry $\theta:L^q(0,T;X')\longrightarrow \Big(L^p(0,T;X)\Big)'$.

In particular, taking $X=H_0^1(\Omega)$, we conclude that the dual of $L^2(0,T;H_0^1(\Omega))$ is $L^2(0,T;H^{-1}(\Omega))$.

Remark: the explicit form of $f$ allows us to characterize the weak convergence in the dual (which is done in this other post).