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A Ramsey Theorem with an Application to Sequences in Banach Spaces |
Canad. Math. Bull. 55(2012), 410-417
Published:2011-04-15
Printed: Jun 2012
Printed: Jun 2012
- Robert Service,
Abstract
The 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.
| Keywords: | Banach spaces, Ramsey theory |
| MSC Classifications: |
46B15 - Summability and bases [See also 46A35] 05D10 - Ramsey theory [See also 05C55] |