It is known that $[L^p(0,T;H)]^* = L^q(0,T;H^*)$.

If $p=q=2$ and $H$ is a Hilbert space, is there an easier proof to show that the spaces are **isometric**? The proof that I know for the general case uses some sort of epsilon argument, but there must be an easier way when we have access to Riesz maps?