Let $V=L^2(0,T;H^1) + L^p(0,T;L^p)$. We know that its dual space is $V^* = L^2(0,T;H^{-1}) \cap L^p(0,T;L^p)$.
So if $v \in V$, then by definition $v=a+b$ where $a \in L^2(0,T;H^1)$ and $b \in L^p(0,T;L^p)$.
My question is, given $f \in V^*$, what does a duality pairing $$\langle f, v \rangle_{V^*, V}$$ look like? Is it composed of the duality pairings of $L^2(0,T;H^{-1})$ and $L^{p'}(0,T;L^{p'})$ in some way? How to think of it intuitively?
If $p =2$, then isn't $V^*=L^2(0,T;L^2)$? But then the dual of $V^*$, which should be $V$, is $L^2(0,T;L^2)$ too, but this is not the same space as $L^2(0,T;H^1) + L^2(0,T;L^2)$!!