Almost Over Complete and Over Total Sequences with Polynomials and Complex Structures on Banach Space

Abstract

We introduce an almost over complete sequence in a Banach space and almost overtotal sequence in a dual space. We show that any of such sequences is relatively norm- compact. We study Banach space of traces of real polynomials on the Euclidean space to a compact subsets equipped with supremum norms. We develop a notion of a dimension where a Banach space with a uniformly bounded action of sofic group is a sofic approximation. We also develop a notion of the dimension with an embeddable group and the space of finite- dimensional Schatten p- class operators. We give examples of real Banach spaces with exactly infinite countably many complex structures.