Antonio Martínez-Abejón

ULTRAPOWERS AND SUBSPACES OF THE DUAL OF A BANACH SPACE

For a Banach space X we consider three ways in which a subspace of X∗ can represent locally the whole dual space X∗. We obtain characterizations in terms of ultrapowers and we study the relationship between the subspaces of X∗ and the subspaces of the dual of an ultrapower of X . 2000 Mathematics Subject Classification. Primary: 46B08, 46B10. Secondary: 46B04, 46B20.

Local dual spaces of Banach spaces of vector-valued functions

We show that L∞ (μ, X* ) is a local dual of L 1 (μ,X ), and L 1 (μ, X*) is a local dual of L∞ (μ, X), where X is a Banach space. A local dual space of a Banach space Y is a subspace Z of Y* so that we have a local representation of Y* in Z satisfying the properties of the representation of X** in X provided by the principle of local reflexivity.

Ultrapowers ofL 1(μ) and the subsequence splitting principle

AbstractWe study the ultrapowers

Finite representability of operators

L_1 (\mu )_\mathfrak{U}

Tauberian operators. Basic properties

of aL1(μ) space, by describing the components of the well-known representation

Tauberian operators on spaces of integrable functions

L_1 (\mu )_\mathfrak{U} = L_1 (\mu _\mathfrak{U} ) \oplus _1 L_1 (\nu _\mathfrak{U} )