2020/09/07 by S. Garcı́a-Ferreira, Garcia-Ferreira, S., A. C. Hernandez-Soto +1
Mathematics · #05C55 #05D10 #46B15 #Advanced Banach Space Theory #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.2009.03247
openalex publication_date 2020/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A well-known application of the Ramsey Theorem in the Banach Space Theory is the proof of the fact that every normalized basic sequence has a subsequence which generates a spreading model (the Brunel-Sucheston Theorem). Based on this application, as an intermediate step, we can talk about the notion of (k,ε)-oscillation stable sequence, which will be described and analyzed more generally in this article. Indeed, we introduce the notion ((Bi)i=1k,ε)-block oscillation stable sequence where (Bi)i=1k is a finite sequence of barriers and using what we will call blocks of barriers. In particular, we prove that the Ramsey Theorem is equivalent to the statement ``for every finite sequence (Bi)i=1k of barriers, every ε>0 and every normalized sequence (xi)i∈ℕ there is a subsequence (xi)i∈ M that is ((Bi\capP(M))i=1k,ε)-block oscillation stable'', where P(M) is the power set of the infinite set M. Besides, we introduce the (Bi)i∈ℕ-block asymptotic model of a normalized basic sequence where (Bi)i∈ℕ is a sequence of barriers. These models are a generalization of the spreading models and are related to the ((Bi)i=1k,ε)-block oscillation stable sequences. We show that the Brunel-Sucheston is satisfied for the (Bi)i∈ℕ-block asymptotic models, and we also prove that this result is equivalent to the Ramsey Theorem. The difference between our theorem and the Brunel-Sucheston Theorem is based on the number of different models that are obtained from the same normalized basic sequence through them. This and other observations about (Bi)i∈ℕ-block asymptotic models are noted in an example at the end of the article.