ADVANCED PROPERTIES OF LEBESGUE – RIESZ AND BANACH SPACES WITH ALMOST TRANSITIVE NORM

Abstract

It is shown that Banach space of certain density contains an unconditional sequence of cardinality and a separable super reflexive Banach space admits a continuous surjection on each of its subspaces. We introduce a new class of Banach spaces that are called sub B- convex and show that any separable sub B- convex Banach space may be almost isometrically embedded in a separable Banach space of the same cotype. We show a method for producing Banach spaces with the Tsirelson property and a sub symmetric basis which enjoys the hereditarily approximative property. We obtained a result on the existence of an uncomplemented subspace isomorphic to the Hilbert space and proved that for any pair of symmetric spaces there exists an intermediate space that is reflexive or weakly sequentially complete. We study the isometric version of the amalgamation property in Banach spaces.