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SOME PROPERTIES OF BOCHNER INTEGRAL IN BITOPOLOGICAL VECTOR SPACES AND INTRODUCTION TO GENERALIZED LEBESGUE SPACES $L^p(E,(X_\vartheta,\|.\|))$

S. Lahrech, A. Ouahab, A. Benbrik, A. Mbarki

Abstract


We consider a bitopological vector space $(X,\vartheta,\|.\|)$, where $(X,\vartheta)$ is a topological vector space, and $\|.\|$ is a norm defined on $X$. We give some properties of the Bochner integral with respect to the pair of topologies $(\vartheta,\|.\|)$, and we introduce a special class of integrable functions denoted $L^p(E,(X_{\vartheta},\|.\|))$, which contains the usual Lebesgue space $L^p(E,(X,\|.\|))$. Next, we give an example which shows that the canonical injection of\linebreak  $L^p(E,(X,\|.\|))$ into $L^p(E,(X_{\vartheta},\|.\|))$  is in general strict.

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