Majorizing measures and their applications

Abstract

Majorizing measure techniques are developed and applied to Banach space theory. In particular, the following is proved.

Let B₁ and B₂ be the unit balls of [special characters omitted] and [special characters omitted], respectively, relative to the canonical basis [special characters omitted]. Suppose [special characters omitted]. Then for every [special characters omitted] > 0, there exist [special characters omitted] with cardinality [special characters omitted], and constant C depending only on [special characters omitted] and p, such that [special characters omitted], where [special characters omitted] is the linear span of [special characters omitted].

The following is a consequence.

Consider vectors [special characters omitted] in the unit ball of a Banach space X, and [special characters omitted] If X is of type 2 and X* is uniformly convex, then, there exists a constant C depending only on [special characters omitted] and [special characters omitted], such that for a randomly selected subset I of cardinality [special characters omitted], [special characters omitted] for all scalar sequence [special characters omitted].

This solves a problem stated in [T2].