Type and Cotype of Normed Spaces, and Some Simple Relations with Geometrical Properties

Abstract

Let X be a normed space, x _i ∈ X, i = 1,2,... . In this section we will study the averages $\left( {\mathop {Ave}\limits_{\varepsilon _i = \pm 1} \left\| {\sum\nolimits_{i = 1}^n {\varepsilon _i x_i } } \right\|^2 } \right)^{1/2}$ and see that the order of magnitude of these averages gives a lot of information about some geometrical properties of X.