Ramsey Methods in Analysis

1. Front Matter

   Pages i-vii

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2. Saturated and Conditional Structures in Banach Spaces

   1. Introduction

      Pages 3-6

   2. Tsirelson and Mixed Tsirelson Spaces

      Pages 7-19

   3. Tree Complete Extensions of a Ground Norm

      Pages 21-38

   4. Hereditarily Indecomposable Extensions with a Schauder Basis

      Pages 39-46

   5. The Space of the Operators for Hereditarily Indecomposable Banach Spaces

      Pages 47-55

   6. Examples of Hereditarily Indecomposable Extensions

      Pages 57-70

   7. The Space \(\mathfrak{X}\omega _1 \)

      Pages 71-79

   8. The Finite Representability of \(J_{T_0 }\) and the Diagonal Space \(D \left( {\mathfrak{X}_\gamma } \right)\)

      Pages 81-86

   9. The Spaces of Operators \(L\left( {\mathfrak{X}_\gamma } \right)\) , \(L\left( {X,\mathfrak{X}\omega _1 } \right)\)

      Pages 87-98

   10. Transfinite Schauder Basic Sequences

       Pages 99-103

   11. The Proof of the Finite Representability of \(J_{T_0 }\)

       Pages 105-116

   12. Bibliography

       Pages 117-120

3. High-Dimensional Ramsey Theory and Banach Space Geometry

   1. Introduction

      Pages 123-126

   2. Finite-Dimensional Ramsey Theory: Finite Representability of Banach Spaces

      Pages 127-141

   3. Ramsey Theory of Finite and Infinite Sequences

      Pages 143-196

   4. Ramsey Theory of Finite and Infinite Block Sequences

      Pages 197-216

   5. Approximate and Strategic Ramsey Theory of Banach Spaces

      Pages 217-246

   6. Bibliography

      Pages 247-252

4. Back Matter

   Pages 253-257

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This book contains two sets of notes prepared for the Advanced Course on R- sey Methods in Analysis given at the Centre de Recerca Matem` atica in January 2004, as part of its year-long research programme on Set Theory and its Appli- tions. The common goal of the two sets of notes is to help young mathematicians enter a very active area of research lying on the borderline between analysis and combinatorics. The solution of the distortion problem for the Hilbert space, the unconditional basic sequence problem for Banach spaces, and the Banach ho- geneous space problem are samples of the most important recent advances in this area, and our two sets of notes will give some account of this. But our main goal was to try to expose the general principles and methods that lie hidden behind and are most likely useful for further developments. The goal of the ?rst set of notes is to describe a general method of building norms with desired properties, a method that is clearly relevant when testing any sort of intuition about the in?nite-dimensional geometry of Banach spaces. The goal of the second set of notes is to expose Ramsey-theoretic methods relevant for describing the rough structure present in this sort of geometry. We would like to thank the coordinator of the Advanced Course, Joan Ba- ria, and the director of the CRM, Manuel Castellet, for giving us this challenging but rewarding opportunity. Part A SaturatedandConditional StructuresinBanachSpaces SpirosA.

• Banach Space
• Banach spaces
• Finite
• Ramsey method
• Ramsey theory
• calculus
• geometry
• proof
• combinatorics

"This book is the result of lectures given by the authors aimed at bringing young researchers to the forefront of a ‘very active research area lying on the borderline between analysis and combinatorics’…This book will certainly be appreciated by experts. It is also valuable for young researchers who are suitably prepared and wish to work in this amazing area."   —Mathematical Reviews

"The book is carefully written with clear exposition of the material. It can be studied by graduate students who had a first course in functional analysis and are interested in either functional analysis or Ramsey theory."   —Zentralblatt MATH

• Department of Mathematics, National Technical University, Greece

  Spiros A. Argyros

• U.M.R. 7057, Université Paris 7 — C.N.R.S., Paris Cedex 05, France

  Stevo Todorcevic

• Department of Mathematis, University of Toronto, Toronto, Canada

  Stevo Todorcevic

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