Abstract.
In [6] W. T. Gowers formulated and proved a Ramsey-type result which lies at the heart of his famous dichotomy for Banach spaces. He defines the notion of weakly Ramsey set of block sequences of an infinite dimensional Banach space and shows that every analytic set of block sequences is weakly Ramsey. We show here that Gowers’ result follows quite directly from the fact that all Gδ sets are weakly Ramsey, if the Banach space does not contain c0, and from the fact that all F σδ sets are weakly Ramsey, in the case of an arbitrary Banach space. We also show that every result obtained by the application of Gowers’ theorem to an analytic set can also be obtained by applying the Theorem to a F σδ set (or to a G δ set if the space does not contain c0). This fact explains why the only known applications of this technique are based on very low-ranked Borel sets (open, closed, F σ , or G δ ).
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