GEOMETRIC AND ISOMETRIC PROPERTIES OF TWO CLASSES OF SEQUENCE SPACES

We study geometric and isometric properties of two classes of sequence spaces. First, we consider the Banach sequence space Xn, the com-pletion of c00 relatively to the norm llxlln = sup ⎧ ⎨ ⎩ � i & ISIN;F |xi|, with , F of cardinality at most n ⎫ ⎬ ⎭ . For this space, we show that the isometry group of Xn consists of standard isometries. Second, we consider a Tsirelson-type space, T [& theta;, An] (0 < & theta; < 1) associated with the regular family An, comprised of all subsets of N with at most n elements. For this space, we characterize the extreme points and show that for & theta; & LE; n1, T [& theta;, An] supports standard isometries. We also derive the form for the surjective isometries of T[& theta;, A3].