Some Structural Properties of Banach Spaces

Abstract

In this paper, we describe briefly some structural properties of Banach space. We organize this paper into three parts. First, we investigate several long James-type Banach spaces and get representation theorems of long James-type Banach spaces and their duals by using transfinite basis. Second, we use Edgar ordering to consider sums of Banach spaces, sequence Banach spaces, function Banach spaces and compare them with C 0, l p , James space, long James space and so on. We find out the long James space J(ω 1 ) is a predecessor of the continuous function space C[0,ω 1 ] in this ordering,however J(ω 1 ) and C[0,ω 1 ] both are not predecessors of l ∞ .So we find out the existence of new ordering chains on the class of Banach spaces. We also describe the ordering structure about vector valued long James spaces J(η,X),J(η,l p )(1 <p< ∞) and we get J(η,l p ) = J(η)(1 <p< ∞) and J(η)l p ) < J(η),l ∞ ,) and some others. Finally, we introduce and use some geometric constants of Banach space,for example R α (X), to investigate the geometric properties of Banach spaces.