Dense subspaces of $L^p(0,T;X)$

Question

Given a Banach space $X$ and $1\leq p<\infty$, let's define the space $L^p(0,T;X)$ as the set of all strongly measurable functions $f:(0,T)\mapsto X$ such that $$\int_0^T\Vert f\Vert_{X}^pdt<\infty$$ (with the usual modifications when $p=\infty$). I'm looking for dense subspaces of $L^p(0,T;X)$; for instance, we konw that for usual Lebesgue space $C_0^\infty$ is dense in $L^p$. Are there similar results for the space $L^p(0,T;X)$? In particular I would like to regularize a function in $L^p(0,T;X)$ in time.

Answer 1

Unless you use nonstandard definitions, the desired density holds, and for the same reason as in the scalar case: Simple functions are dense in $L_p(0,T:X)$, after that you can use the same argument as in the scalar case (multiplying the smooth functions approximating indicator functions of sets by the corresponding vectors). As for the density, you can find it in Section 1.1 of these notes by Pisier (a substantially expanded version of these notes is scheduled for publication in Cambridge University Press in 2016).