#### 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 B*_{1} *and B*_{2} * be the unit balls of* l^{n}1 *and* l^{n}2 , *respectively, relative to the canonical basis* ei ni=1 . *Suppose* K⊂log^{p} nB1∩B 2 . *Then for every* 3 > 0, *there exist* S⊂1,2,&ldots;,n *with cardinality* n^{1-3} , *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=n^{1-3}/s , i∈I aixi≤C i∈I ai^{2}^{ 1/2}, *for all scalar sequence* ai ni=1 . ^ This solves a problem stated in [T2]. ^

#### Recommended Citation

Gao, Fuchang, "Majorizing measures and their applications" (1999). *Doctoral Dissertations*. AAI9942573.

https://opencommons.uconn.edu/dissertations/AAI9942573