TITLE: Is $L^{p}(I,X)\cong L^{p}(I) \widehat{\otimes_{\pi}}X$?
QUESTION [6 upvotes]: It is well known that
$$L^{1}(I,X) \cong L^{1}(I) \widehat{\otimes_{\pi}}X$$
For any compact interval (I,m) with the Lebesgue measure and any Banach space X.
Is it still true when $1<p<\infty$
that
$$L^{p}(I,X) \cong L^{p}(I) \widehat{\otimes_{\pi}}X$$
If not, can we embed one of them in the other ?

REPLY [3 votes]: If by $\otimes_\pi$ you mean the projective tensor product then the answer is positive for $p=1$ as you remarked and otherwise it is false. Indeed, take $p=2$ and $X=L_2$. Then $L_2(L_2)$ is a Hilbert space but $L_2\otimes_\pi L_2$ is the dual of the space of compact operators on $L_2$, hence it is non-reflexive.