TOPOLOGICAL PROPERTIES AND EXTENSION OF OPERATORS IN CLASSES OF FINITELY EQUIVALENT BANACH SPACES

Abstract

We show the classes of Banach spaces that are finitely, strongly finitely or elementary equivalent. On sets of these classes topological properties are defined in such away that sets of defined classes become compact totally disconnected topological spaces. Result are used in the problem of synthesis of Banach spaces, and to describe omittable spaces that have been defined. We considered two problems on extension of operators whose range spaces for the first problem (or domain spaces for the second one) belongs to the fixed class of finite equivalence, which is generated by a given Banach space X. Both problems are considered in two variables: isometric and isomorphic ones. The first problem was solved completely, while a solution was obtained for the second problem which is close to the exact solution.