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

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 $L^p(E,(X_{\vartheta},\|.\|))$.