Approximate and Strategic Ramsey Theory of Banach Spaces

Errata and addendum: tensor products and Dunford–Pettis sets

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004107000102 ◽

2007 ◽

Vol 143 (2) ◽

pp. 387-390

Author(s):

Ioana Ghenciu ◽

Paul Lewis

Keyword(s):

Banach Spaces ◽

Ramsey Theory ◽

Tensor Products ◽

Schauder Basis ◽

Fundamental Result ◽

Individual Basis ◽

Quick Proof

AbstractGhenciu and Lewis introduced the notion of a strong Dunford–Pettis set and used this notion to study the presence or absence of isomorphic copies of c0 in Banach spaces. The authors asserted that they could obtain a fundamental result of J. Elton without resorting to Ramsey theory. While the stated theorems are correct, unfortunately there is a flaw in the proof of the first theorem in the paper which also affects subsequent corollaries and theorems. The difficulty is discussed, and Elton's results are employed to establish a Schauder basis proposition which leads to a quick proof of the theorem in question. Additional results where questions arise are discussed on an individual basis.

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A Ramsey Theorem with an Application to Sequences in Banach Spaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-073-5 ◽

2012 ◽

Vol 55 (2) ◽

pp. 410-417

Author(s):

Robert Service

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Ramsey Theory ◽

Basic Sequence ◽

Alternative Proof ◽

Ramsey Theorem ◽

Combinatorial Lemma

AbstractThe notion of a maximally conditional sequence is introduced for sequences in a Banach space. It is then proved using Ramsey theory that every basic sequence in a Banach space has a subsequence which is either an unconditional basic sequence or a maximally conditional sequence. An apparently novel, purely combinatorial lemma in the spirit of Galvin's theorem is used in the proof. An alternative proof of the dichotomy result for sequences in Banach spaces is also sketched, using the Galvin–Prikry theorem.

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