I'm trying to find a good reference for the claim from the caption. In my case, I have $V = H_0^1(\Omega)$ and $H = L^2(\Omega)$. I already know (or am pretty sure) that $C([0,T];V)$ is dense in $L^2((0,T);V)$ (since a similar result holds in usual Lp spaces) and that $L^2((0,T);V)$ is dense in $L^2((0,T);H)$ but I don't really have any reference for that.
I'm actually not discussing Bochner spaces in detail in my thesis but still need this result, i.e. a good reference for it. Any help is appreciated! Thanks in advance :)