Title

Majorizing measures and their applications

Date of Completion

January 1999

Keywords

Mathematics

Degree

Ph.D.

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 ln1 and ln2 , respectively, relative to the canonical basis ei ni=1 . Suppose K⊂logp nB1∩B 2 . Then for every 3 > 0, there exist S⊂1,2,&ldots;,n with cardinality n1-3 , and constant C depending only on 3 and p, such that K∩YS⊂CB1∩YS , where YS is the linear span of eii∈S . ^ The following is a consequence. ^ Consider vectors x1,x2,&ldots;,xn in the unit ball of a Banach space X, and s=supi≤ nx* xi 2;x*∈X*,∥ x* ∥ ≤1. If X is of type 2 and X* is uniformly convex, then, there exists a constant C depending only on 3 and T2X , such that for a randomly selected subset I of cardinality m=n1-3/s , i∈I aixi≤C i∈I ai2 1/2, for all scalar sequence ai ni=1 . ^ This solves a problem stated in [T2]. ^