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Duality of Bochner spaces. (arXiv:1801.09059v2 [math.FA] UPDATED)
来源于:arXiv
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. 查看全文>>