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 May 30 revised $H^1(\Omega) \subset L^2(\Omega)$ dense for $\Omega$ a $C^l$ hypersurface with boundary? edited body May 30 asked $H^1(\Omega) \subset L^2(\Omega)$ dense for $\Omega$ a $C^l$ hypersurface with boundary? May 24 awarded Scholar May 18 comment Proof that $L^2(0,T;X)^* = L^2(0,T;X^*)$ @AndrasBatkai Is it as simple as: $L^2(0,T;X)' = L^2(0,T;X) = L^2(0,T;X')$? Where the first equality is by Riesz representation theorem (RRT) for the Bochner space (which is Hilbert), and the second is by RRT for $X$ which is also Hilbert. Would this be a good proof?? (The longer way to do this requires us to show that the map is isometric, which seems difficult to do even in this case.) May 17 accepted Proof that $L^2(0,T;X)^* = L^2(0,T;X^*)$ May 17 comment Proof that $L^2(0,T;X)^* = L^2(0,T;X^*)$ @Andras you are right. But I am interested in more general $X$, but your answers caters for this. Thanks. May 16 awarded Student May 16 awarded Editor May 16 comment Proof that $L^2(0,T;X)^* = L^2(0,T;X^*)$ @Wlodzimierz I added some details. These are Bochner spaces. May 16 revised Proof that $L^2(0,T;X)^* = L^2(0,T;X^*)$ added 102 characters in body; added 4 characters in body; deleted 2 characters in body May 16 asked Proof that $L^2(0,T;X)^* = L^2(0,T;X^*)$