Ramsey methods in analysis
type: Item
instance of: scholarly article
title: Ramsey methods in analysis
zbMATH Open document ID: 1092.46002
author: Spiros A. Argyros
author: Stevo Todorčević
published in: Advanced Courses in Mathematics -- CRM Barcelona
publication date: 2005-06-08T00:00:00Z
review text: The book under review has two parts. The first part ``Saturated and unconditional structures in Banach spaces'' is written by Spiros A. Argyros and presents constructions of mixed Tsirelson and hereditarily indecomposable (HI) spaces. The second part ``High-dimensional Ramsey theory and Banach space geometry'' is written by Stevo Todorcevic and presents connections between Ramsey theory and Banach space geometry. The first part of the book has eight chapters. Chapter~I discusses Tsirelson spaces and mixed Tsirelson spaces. Tsirelson space was the first known example of a Banach space that does not contain any isomorph of \(\ell_p\) (\(1 \leq p < \infty\)) or \(c_0\). The mixed Tsirelson spaces provide a generalization of the classical Tsirelson space and they were introduced and studied by Argyros and Deliyanni. Chapter~II presents the mixed Tsirelson extensions of a ground norm. These were introduced by Argyros and Tolias. Starting from a set \(G\) (which satisfies very mild assumptions and is called a \textit{ground set}) which is a subset of \(c_{00} (\kappa )\), the finitely supported sequences on an ordinal \(\kappa\), one defines a norm \(\| { \cdot \| _G}\) on \(c_{00} (\kappa )\) so that \(G\) is a norming set and denotes by \(Y_G\) its completion. The norm of the mixed Tsirelson extension \(T_{\kappa}[G]\) of \(Y_G\) is defined as an increasing limit of a sequence of norms which are defined inductively having \({\| \cdot \| _G}\) as the starting point. If \(T_{\kappa}[G]\) is a strictly singular extension of \(Y_G\) (i.e., if the identity map from \(T_{\kappa}[G]\) to \(Y_G\) is strictly singular) then \(T_{\kappa}[G]\) is reflexive and unconditionally saturated. Moreover \(T_{\kappa}[G]\) does not contain any \(\ell_p\) (\(1 \leq p < \infty\)). The transfinite basis of \(T_{\kappa}[G]\) is boundedly complete, hence \(T_{\kappa}[G]\) is a dual space. The tree complete extension \({{\mathfrak X}} [ G, \sigma]\) of \(Y_G\) was introduced by Argyros and Tolias…
Mathematics Subject Classification ID: 46-02
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 05D10
zbMATH DE Number: 2174318
zbMATH Keywords: mixed Tsirelson spaces
zbMATH Keywords: hereditarily indecomposable Banach spaces
zbMATH Keywords: Ramsey theory
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2174318