Type: Article

Ramsey property and block oscillation stability on normalized sequences in Banach spaces

Hernández-Soto, A. C., García-Ferreira, S.;

Abstract:

A well-known application of the Ramsey Theorem is the proof of the Brunel–Sucheston Theorem. Based on this application, as an intermediate step, we consider the concept of (k, ?) -oscillation stable sequence, which is generalized with the notion of ((Bi)i=1k,?)-block oscillation stable sequence where (Bi)i=1k is a finite sequence of barriers on N. We prove that the Ramsey Theorem is equivalent (i.e., one result is deduced by using the other) to the statement:“for every finite sequence(Bi)i=1kof barriers, every?> 0 and every normalized sequence(xi)i?Nthere exists a subsequence(xi)i?Mthat is((Bi?P(M))i=1k,?)-block oscillation stable”, where P(M) is the power set of the infinite set M. We also prove a theorem like the Brunel–Sucheston Theorem which introduces the notion of (Bi)i?N-block asymptotic model of a normalized basic sequence where (Bi)i?N is a sequence of barriers. This notion includes the notion of spreading model of Brunel–Sucheston and we provide an example of a block asymptotic model that is not a spreading model. Moreover, we observe that some of our main results are equivalent to the Ramsey Theorem