2023/03/07 by Quentin Menet, Menet, Quentin, Dimitris Papathanasiou +1
Computer Science · Mathematics · #15A03 #46B87 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Differential Equations Analysis #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2303.03871
openalex publication_date 2023/03/07 · openalex created_date 2023/03/10 · openalex updated_date 2026/07/28
For each vector x∈ ℓ∞, we can define the non-empty compact set Lx of accumulation points of x. Given an infinite subset A of ℕ\backslash\1\, we can therefore investigate under which conditions on A, the set L(A):=\x∈ ℓ^∞: |Lx|∈ A\ is lineable or even densely lineable. In particular, we show that if L(A) is lineable then there exists k≥ 1 such that A∩ (A-k) is infinite and that if L(A) is densely lineable then A∩ (A-1) is infinite. We end up by answering an open question on the existence of a closed non-separable subspace in which each non-zero vector has countably many accumulation points.