Geometric and isometric properties of two classes of sequence spaces

Abstract

We study geometric and isometric properties of two classes of sequence spaces. First, we consider the Banach sequence space X n X_n , the completion of c 00 c_{00} relatively to the norm \[ ‖ x ‖ n = sup { ∑ i ∈ F | x i | , with , F of cardinality at most n } . \|x\|_n = \sup \left \{\sum _{i \in F}|x_i|, \text { with }, F \text { of cardinality at most } n\right \}. \] For this space, we show that the isometry group of X n X_n consists of standard isometries. Second, we consider a Tsirelson-type space, T [ θ , A n ] T[\theta , A_n] ( 0 > θ > 1 0>\theta >1 ) associated with the regular family A n A_n , comprised of all subsets of N \mathbb {N} with at most n n elements. For this space, we characterize the extreme points and show that for θ ≤ 1 n \theta \leq \frac {1}{n} , T [ θ , A n ] T[\theta , A_n] supports standard isometries. We also derive the form for the surjective isometries of T [ θ , A 3 ] T[\theta , A_3] .