User:ByVarying/sandbox

Examples

 * For $$(X, \mathcal{A}, \mu)$$ a measure space, let $$L^1$$ be the Lebesgue space $$L^1(\mu)$$. Let $$E$$ be a Banach space over $$\Reals$$. Let $$L^1_E$$ be the completion of the space of simple functions $$X\to E$$, modulo the subspace of functions $$X\to E$$ whose pointwise norms, considered as functions $$X\to\Reals$$, have integral $$0$$ with respect to $$\mu$$. Then $$L^1_E$$ is isometrically isomorphic to $$L^1 \widehat{\otimes}_\pi E$$.