The duality properties of the integration map associated with a vector measure m are used to obtain a representation of the (pre)dual space of the space L p (m) of p-integrable functions (where 1 < p < ∞) with respect to the measure m. For this, we provide suitable topologies for the tensor product of the space of q-integrable functions with respect to m (where p and q are conjugate real numbers) and the dual of the Banach space where m takes its values. Our main result asserts that under the assumption of compactness of the unit ball with respect to a particular topology, the space L p (m) can be written as the dual of a suitable normed space.2000 Mathematics subject classification: primary 46G10; secondary 46B28.
We study continuity and other properties related to some kind of compactness of multiplication operators between different spaces of pth power integrable scalar functions with respect to a vector measure.
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