TENSOR PRODUCT REPRESENTATION OF THE (PRE)DUAL OF THE <i>L</i>^<i>p</i>-SPACE OF A VECTOR MEASURE

Abstract

Abstract 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&lt; p &lt; ∞ ) 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.