# Dual space of Bochner space: is there an easier proof to show they're isometric?

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?

-
–  András Bátkai May 20 '13 at 19:45
Uhm, I might be underestimating the problem, but I believe that if $H$ is a Hilbert space, then so is $L^2(0,T;H)$. Isometry then follows directly from Riesz Representation Theorem.