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

Question

I have two questions, it will be acknowledged if someone can give me some help.

1. If $\Omega\subset R^n$ a bounded domain, what the dual of $L^2(0, T; H_0^1(\Omega))$? Is that $L^2(0, T; H^{-1}(\Omega))$?

2. When it holds that $$H_0^1(\Omega)\subseteq L^2(\Omega)=(L^2(\Omega))^*\subseteq H^{-1}(\Omega),$$ how to show $$(f, g)_{(H^{-1}, H_0^1)}=(f, g)_{(L^2,L^2)}?$$ Thanks a lot!