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ON THE COMPLETENESS, SEPARABILITY AND DENSITY THEOREMS FOR GENERALIZED LEBESGUE-BOCHNER SPACE $L^p(E,(X_\vartheta,\|.\|))$

S. Lahrech A. Ouahab, A. Benbrik, A. Mbarki, Islam Eddine Hadi

Abstract


Recently, S. Lahrech and al (see \cite{r0}) have introduced a special class of
integrable functions denoted $L^p(E,(X_{\vartheta},\|.\|))$ which contains the usual
Lebesgue-Bochner space $L^p(E,(X,\|.\|))$, where $(X,\vartheta)$ is a topological vector
space, and $\|.\|$ is a norm defined on $X$. Many properties of $L^p(E,(X_{\vartheta},\|.\|))$
have been established in \cite{r0}.

The purpose of this paper is to continue the study of the
generalized Lebesgue-Bochner space $L^p(E,(X_{\vartheta},\|.\|))$
in the case where the unit ball ${\cal B}_1(X)$ is closed in
$(X,\vartheta)$ and sequentially complete under the topology
$\vartheta$. Under the above conditions, we establish some results
related to the separability and completeness of
$L^p(E,(X_{\vartheta},\|.\|))$. Moreover, we prove that the class
${\cal C}(E,(X_\vartheta,\|.\|))$ of continuous vector functions
from $E$ into $X_{\vartheta}$ and bounded with respect to the
topology generated by the norm $\|.\|$ is dense in
$L^p(E,(X_{\vartheta},\|.\|))$.

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