2021/07/25 by Nizar El Idrissi, Idrissi, Nizar El, Samir Kabbaj +1 · 1 citation
Mathematics · #03C07 #06A12 #15A03 #15A04 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2107.11834
openalex publication_date 2021/07/25 · openalex created_date 2022/07/24 · openalex updated_date 2026/07/28
In [Appl. Comput. Harmon. Anal., 46(3):664-673, 2019], O. Christensen and M. Hasannasab observed that assuming the existence of an operator T sending en to en+1 for all n ∈ ℕ (where (en)n ∈ ℕ is a sequence of vectors) guarantees that (en)n ∈ ℕ is linearly independent if and only if dim(span\en\n ∈ ℕ) = ∞. In this article, we recover this result as a particular case of a general order-theory-based model-theoretic result. We then return to the context of vector spaces to show that, if we want to use a condition like T(ei)=eϕ(i) for all i ∈ I where I is countable as a replacement of the previous one, the conclusion will only stay true if ϕ: I → I is conjugate to the successor function succ : n ↦ n+1 defined on ℕ. We finally prove a tentative generalization of the result, where we replace the condition T(ei)=eϕ(i) for all i ∈ I where ϕ is conjugate to the successor function with a more sophisticated one, and to which we have not managed to find a new application yet.