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ON THE TOPOLOGICAL DUAL OF GENERALIZED LEBESGUE-BOCHNER SPACE $L^p(E,(X_\vartheta,\|.\|))$

S. Lahrech, A. Ouahab, A. Benbrik, A. Mbarki, I.E. Hadi

Abstract


Let $(X,\|.\|,\vartheta)$ be a bitopological vector space such that
$(X,\vartheta)$ is a topological vector space, $(X,\|.\|)$ is a reflexive normed space,
the unit ball ${\cal B}_1(X)$ is closed in
$(X,\vartheta)$ and sequentially complete under the topology $\vartheta$.

Let $p$ and $q$ be such that $1\leq p<\infty$, $1<q\leq \infty$ and $\frac{1}{p}$
+$\frac{1}{q}=1$. Denote by $l(X_\vartheta,R)$ the space of all sequentially
continuous linear mapping from  $(X,\vartheta)$ to $R$. Assume that every bounded
subset of $(X,\|.\|)$ is bounded in $(X,\vartheta)$. In this case, $l(X_\vartheta,R)$
is contained in $(X,\|.\|)'$.\

The aim of this article is to generalize the result related to the topological dual
of Lebesgue-Bochner space $L^p(E,(X,\|.\|))$ to generalized
Lebesgue-Bochner space $L^p(E,(X_{\vartheta},\|.\|))$
(see \cite{r1} for more details about the notion of generalized Lebesgue-Bochner spaces).
Thus, we prove that the topological dual of generalized Lebesgue-Bochner space
$L^p(E,(X_{\vartheta},\|.\|))$ can be identified algebraically and topologically with \pagebreak
the Lebesgue-Bochner space $L^q(E,(l(X_\vartheta,R),\|.\|_{X'}))$. In particular, if
the topology $\vartheta$ coincides with the topology generated by the norm $\|.\|$, then
one finds the classical result $(L^p(E,(X,\|.\|)))'\simeq L^q(E,(X',\|.\|_{X'}))$.

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