Probability and Measure

The goal of this chapter is to reveal the power of descriptive set theory in penetrating the structure of Banach spaces. The chapter is divided into three subchapters, each with its own introduction. Subchapters one, two and three were mostly written by the second, third, and first authors, respectively. Space limitations forced us to leave out many fundamental results and on-going research in this exciting interface. We do hope, however, that our article gives a strong flavor of this aspect of Banach space theory.

Descriptive Set Theory and Banach Spaces

It is proved that every normalized weakly null \sq\ has a sub\sq\ which is convexly unconditional. Further, an Hierarchy of summability methods is introduced and with this we give a complete classification of the complexity of weakly null \sq s.

Convex unconditionality and summability of weakly null sequences

Genericity and amalgamation of classes of Banach spaces

Review: J. D. Halpern, H. Lauchli, A Partition Theorem; J. D. Halpern, A. Levy, The Boolean Prime Ideal Theorem Does Not Imply the Axiom of Choice

It is shown that for every separable Banach space X with non-separable dual, the space \(X^{**}\) contains an unconditional family of size \(|X^{**}|\) . The proof is based on Ramsey Theory for trees and finite products of perfect sets of reals. Among its consequences, it is proved that every dual Banach space has a separable quotient.

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Unconditional families in Banach spaces

* Basic concepts* Coloring theory: Combinatorial spaces* Strong subtrees* Variable words* Finite sets of words* Density theory: Szemeredi's regularity method* The removal lemma* The density Hales-Jewett theorem* The density Carlson-Simpson theorem* Appendices: Primitive recursive functions* Ramsey's theorem* The Baire property* Ultrafilters* Probabilistic background* Open problems* Bibliography* Index

Ramsey Theory for Product Spaces

The present work consists of three parts. In the first one we determine the prototypes of separable Rosenthal compacta and we provide a classification theorem. The second part concerns an extension of a theorem of S. Todorcevic. The last one is devoted to applications.

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A classification of separable Rosenthal compacta and its applications

Finite order spreading models

We give a purely combinatorial proof of the density Hales-Jewett Theorem that is modeled after the polymath proof but is significantly simpler. In particular, we avoid the use of the equal-slices measure and work exclusively with the uniform measure.

Vassilis Kanellopoulos

Papers

Unconditional families in Banach spaces

Ramsey Theory for Product Spaces

A classification of separable Rosenthal compacta and its applications

Finite order spreading models

A Simple Proof of the Density Hales–Jewett Theorem