INVESTIGATION OF SOME PROBLEMS OF FUNCTIONAL ANALYSIS IN LINEAR SPACES WITH LIMIT OPERATION OF A SEQUENCE

Abstract

This work is based on the theory of spaces with limit operation of a sequence constructed by the author. In a vector space X partially ordered set L of all linear limit operations of a sequence is considered, each of them generates the same system of bounded subsets in X as the given linear limit operation of a sequence. It is proved that L contains the smallest element, each nonempty subset from L has the greatest lower bound and the perfect ordered subset has the least upper bound that is L contains maximal elements. The characteristics of the smallest element and the maximal elements of L are obtained. For linear spaces with limit operation of a sequence statements about a neighborhood of zero, convex sets and differentiable mappings as well as statements that generalize the classical Banach-Steinhaus theorem and the theorem on open mapping are proved. In particular we obtain results reinforcing some known versions of Banach-Steinhaus theorem for topological vector spaces.