On Properties of Symmetric Banach Function spaces and Peetre's Interpolation Spaces

Abstract

A set of all symmetric Banach function spaces defined on [0,1] is equipped with the partial order by the relation of continuous inclusion. Properties of symmetric spaces, which do not depend of their position in the ordered structure, are studied. With the help of the J. Peetre's interpolation scheme it is shown that for any pair of symmetric spaces E, F such that F is absolutely continuously included in E there exists an intermediate (so called, Peetre's) space K(E,F;W), where W is a space with an unconditional basis, that is reflexive or, respectively, weakly sequentially complete provided W also is reflexive or, respectively, weakly sequentially complete.