Abstract
LetS be the binary tree of all sequences of 0’s and 1’s. A chain ofS is any infinite linearly ordered subset. Letℋ be an analytic set of chains, we show that there exists a binary subtreeS’ ofS such that either all chains ofS’ lie inℋ or no chain ofS’ lies inℋ. As an application, we prove the following result on Banach spaces: If (x s) sɛs is a bounded sequence of elements in a Banach spaceE, there exists a subtreeS’ ofS such that for any chainβ ofS’ the sequence (x s ) s ∈β is either a weak Cauchy sequence or equivalent to the usuall 1 basis.
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Stern, J. A ramsey theorem for trees, with an application to Banach spaces. Israel J. Math. 29, 179–188 (1978). https://doi.org/10.1007/BF02762007
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DOI: https://doi.org/10.1007/BF02762007