A note on asymptotically monotone basic sequences and well-separated sets

2019/02/28 by Cleon S. Barroso, Barroso, Cleon S.
Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1902.10857

openalex publication_date 2019/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We remark that if X is an infinite dimensional Banach space then every seminormalized weakly null sequence in X has an asymptotic monotone basic subsequence. We also observe that if X contains an isomorphic copy of ℓ₁, then for every ε>0 there exist a (1 +ε)-equivalent norm \vertiii⋅ on X such that the unit sphere (S_(X, \vertiii⋅)) contains a normalized bimonotone basic sequences which is symmetrically 2-separated.

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