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

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 < 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.

Compactness in spaces of p-integrable functions with respect to a vector measure

“…In this direction, the papers [7,29] show descriptions of the dual space of L p (m), 1 < p < ∞ by characterizing the topology defined by the weak integrals. Σ → E, we write R(m) for its range.…”

“…precise description of (L p (m)) can be found in [7,8,9,10,29]. The integration operator I m : L 1 (m) → E is given by…”

Köthe dual of Banach lattices generated by vector measures

“…The first attempt made use of a topological tensor product formalism for giving a description of these dual spaces ( [9,10,25]) following the original decomposition technique given in [20] for representing the elements of the dual of the space L 1 (ν) as combinations of the elements of the range of ν and some particular integrable functions. However, a representation in terms of Köthe duals of Banach function spaces, i.e.…”

Product spaces generated by bilinear maps and duality

“…This formula concerns the vector measure duality between spaces of p-integrable functions, which was first studied in [14], [15] (see also [7], [8], [16] and the references therein). Roughly speaking, in particular it asserts that the "vector dual" space of L 1 (m) of the vector measure m-i.e., the dual space that appears when the duality is defined by the bilinear operator induced by the integration map-is always L ∞ (m), since the usual dual space of L 1 (m) does not coincide with this space in the general case.…”