On the Banach-Mazur Type for Normed Spaces

TL;DR

This paper introduces a new type for normed spaces to measure the growth of partial sums of weakly null sequences, extending Banach and Mazur's ideas to compare linear dimensions and analyze isomorphisms of classical Banach spaces.

Contribution

It defines a novel type for normed spaces to quantify growth of weakly null sequences, enabling new comparisons of linear dimensions and isomorphisms.

Findings

Introduced a new type for normed spaces based on growth of partial sums.

Compared linear dimensions of classical Banach spaces.

Investigated isomorphisms among classical Banach spaces.

Abstract

In order to measure qualitative properties we introduce a notion of a type for arbitrary normed spaces which measures the worst possible growth of partial sums of sequences weakly converging to zero. The ideas can be traced back to Banach and Mazur who used this type to compare the so-called linear dimension of classical Banach spaces. As an application we compare the linear dimension and investigate isomorphy of some classical Banach spaces.