Reference for: $L^p(S\times\Omega)$ and $L^p(S;L^p(\Omega))$, $p\in[1,\infty)$, are isometric isomorph.

Question

I am having trouble finding a reference for the following result:

Theorem 1. Let $S=(0,T)$ be a finite intervall and $\Omega\subset\mathbb{R}^n$, $n\in\mathbb{N}$, be a bounded domain. Then the spaces $L^p(S\times\Omega)$ and $L^p(S;L^p(\Omega))$, $p\in[1,\infty)$, are isometric isomorph.

The only thing I could find is a proof in Emmrich's book "Gewöhnliche und Operator-Differentialgleichungen" (as far as I know, there is no english translation of this german book). Unfortunately, he only covers the case where $\Omega$ is an one-dimensional interval.

Can anyone point me to a reference in which the general case is proven?

Answer 1

The proof of this fact for genral case can be found in section 7.2 in Tensor norms and operator ideals A. Defant, K. Floret