Duality of Bochner spaces

Abstract

We construct the generalized Lebesgue--Bochner spaces $L^p(\mu,\varPi)$ for positive measures $\mu$ and for suitable real or complex topological vector spaces $\varPi$ so that for $1<p<+\infty$ and Banachable $\varPi$ with separable topology the strong dual of the classical Bochner space $L^p(\mu,\varPi)$ becomes canonically represented by $L^{p^*}(\mu,\varPi_\sigma')\,$. Hence we need no separability assumption of the norm topology of the strong dual $\varPi_\beta'$ of $\varPi$. For $p=1$ and for suitably restricted positive measures $\mu$ we even get a similar result without any separability of the norm topology of the target space $\varPi$. For positive Radon measures on locally compact topological spaces these results are essentially contained on pages 588--606 in R. E. Edwards' classical Functional Analysis.