Reference for: $C_c^\infty(0,T;H)$ dense in $L^2(0,T;H)$

Question

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

Answer 1

$$C^\infty_c(0,T;H) = C^\infty_c(0,T)\otimes^{\ell^1} H = C^\infty_c(0,T)\otimes^{\ell^\infty} H= C^\infty_c(0,T)\otimes^{\ell^2} H$$ 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 Hilbert-Schmidt operators.

Now, since $C^\infty_c(0,T)$ is dense in $L^2(0,T)$, the assertion is obvious.