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Ramsey methods in Banach spaces. (English) Zbl 1043.46018
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland (ISBN 0-444-51305-1/hbk). 1071-1097 (2003).
The author presents an excellent exposition of the interaction of Ramsey theory and Banach spaces during the past 30 years. Among other results, he presents the first application of Ramsey’s original theorem to the notion of a spreading model in a Banach space \(X\): if \((x_i)\) is a normalized basic sequence in \(X\), there exists a subsequence \((y_i)\) so that for all \(n\) and all scalars \((a_i)_1^n\), \(\lim_{k_1\to\infty} \cdots \lim_{k_n\to\infty} \| \sum_{i=1}^n a_i y_{k_i}\| = \| \sum_{i=1}^n a_i e_i\| \) exists. The completion \(E\) of \((\text{span}(e_i),\| \cdot\| )\) is a spreading model of \(X\). He proves the Nash-Williams infinite Ramsey theorem and gives a proof of Rosenthal’s famous \(\ell_1\) theorem: if \((x_i)\) is a bounded sequence in a Banach space, then some subsequence is either weak Cauchy or equivalent to the unit vector basis of \(\ell_1\).
The author also presents proofs of two famous results of his which are Ramsey type theorems. The first says that if \(f\) is a real valued Lipschitz function defined on the unit sphere of \(c_0\) and \(\varepsilon>0\), then there exists an infinite dimensional subspace of \(c_0\) on which \(f\) is constant, up to \(\varepsilon\). The second is his block Ramsey theorem and he deduces his dichotomy theorem: every infinite dimensional Banach space \(X\) contains an infinite dimensional subspace \(Y\) which either has an unconditional basis or is hereditarily indecomposable (if \(Z\) is a subspace of \(Y\) and \(Z = W_1\oplus W_2\), then \(W_1\) or \(W_2\) must be finite dimensional).
The article concludes with the author’s extension of his block Ramsey theorem to infinite block bases and the Banach space consequences of this extension.
For the entire collection see [Zbl 1013.46001].

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B20 Geometry and structure of normed linear spaces
05D10 Ramsey theory
46-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functional analysis