If it is true, where may I find a reference/proof for:
$C_c^\infty(0,T;H)$ dense in $L^2(0,T;H)$
where $H$ is a Hilbert space.
Thanks
If it is true, where may I find a reference/proof for:
where $H$ is a Hilbert space. Thanks 


$$C^\infty_c(0,T;H) = C^\infty_c(0,T)\otimes^{\ell^1} H = C^\infty_c(0,T)\otimes^{\ell^\infty} H= C^\infty_c(0,T)\otimes^{\ell^2} H$$ where I write (for simplicity's sake) $\otimes ^{\ell^1}$ for the completed projective tensor product, also denoted $\pi$tensor product, and $\otimes^{\ell^\infty}$ for the completed injective tensor product, also called $\epsilon$tensor product. The second equality holds, since $C^\infty_c(0,T)$ is a nuclear (LF) space. Thus we have equality also for the $\otimes^{\ell^2}$ tensor product. On the other hand we have $L^2(0,T;H) = L^2(0,T)\otimes^{\ell^2} H$, corresponding to HilbertSchmidt operators. Now, since $C^\infty_c(0,T)$ is dense in $L^2(0,T)$, the assertion is obvious. 

