Let $ B $ be a reflexive, separable Banach space and $ p \in (1,\infty)$. Then denote by $L^{p}(B)$ the space of all functions $$ f : \mathbb{R}^{n} \to B $$ with $$ \int_{\mathbb{R}^{n}} \vert f \vert^{p}_{B} < \infty $$ Now define $L^{p} \times B$ to be the space of all functions $ F : \mathbb{R}^{n} \to B $ such that $F$ is a finite linear combinations of functions of the form $$ f(x) b $$ Where $ f \in L^{p}(\mathbb{R}^{n})$ and $ b \in B $. Then I want a reference that gives a proof to the fact that $L^{p} \times B $ is dense in $L^{p}(B)$. I found the statement written on some book on Harmonic analysis but without a proof and I can't find a reference.
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2$\begingroup$ The space of linear combinations of functions $x\mapsto f(x)b$ is the tensor product $L^p(\mathbb R^n)\otimes B$. Did you check the book Vector Measures of Diestel and Uhl? $\endgroup$– Jochen WengenrothCommented Apr 28 at 6:43
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1$\begingroup$ An explicit reference is Lemma 1.2.19 in T. Hytönen, J. van Neerven, M. Veraar, L. Weis, Analysis in Banach Spaces, Vol. 1 (Springer 2016). Neither reflexivity nor separability matter here. $\endgroup$– Dirk WernerCommented Apr 29 at 19:11
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$\begingroup$ @DirkWerner Thank you for the reference. $\endgroup$– User091099Commented Apr 29 at 19:14
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