期刊名称：International Journal of Differential Equations and Applications
印刷版ISSN：1311-2872
出版年度：2005
卷号：10
期号：2
DOI：10.12732/ijdea.v10i2.190
语种：English
出版社：International Journal of Differential Equations and Applications
摘要：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 $\frac){p}$+$\frac){q}=1$. Denote by $l(X_\vartheta,R)$ the space of all sequentiallycontinuous linear mapping from $(X,\vartheta)$ to $R$. Assume that every boundedsubset 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 dualof Lebesgue-Bochner space $L^p(E,(X,\|.\|))$ to generalizedLebesgue-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 \pagebreakthe Lebesgue-Bochner space $L^q(E,(l(X_\vartheta,R),\|.\|_{X'}))$. In particular, ifthe topology $\vartheta$ coincides with the topology generated by the norm $\|.\|$, thenone finds the classical result $(L^p(E,(X,\|.\|)))'\simeq L^q(E,(X',\|.\|_{X'}))$.
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