<abstract xmlns="http://www.w3.org/1999/xhtml">

In [Appl. Comput. Harmon. Anal., 46 (2019), 664673] O. Christensen and M. Hasannasab observed that assuming the existence of an operator <italic>T</italic> sending <italic>en</italic> to <italic>e</italic><italic>n</italic>+1 for all <italic>n</italic> ∈ ℕ (where (<italic>e</italic><italic>n</italic>)<italic>n</italic>∈ℕ is a sequence of vectors) guarantees that (<italic>e</italic><italic>n</italic>)<italic>n</italic>∈ℕ is linearly independent if and only if dim<italic>{e</italic><italic>n</italic><italic>}</italic><italic>n</italic>∈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 <italic>T</italic>(<italic>e</italic><italic>i</italic>) = <italic>e</italic><italic>ϕ</italic>(<italic>i</italic>) for all <italic>i</italic> ∈ <italic>I</italic> where <italic>I</italic> is countable as a replacement of the previous one, the conclusion will only stay true if <italic>ϕ</italic> : <italic>I</italic> → <italic>I</italic> is conjugate to the successor function <italic>succ</italic> : <italic>n</italic> ↦<italic>n</italic> + 1 defined on ℕ. We finally prove a tentative generalization of the result, where we replace the condition <italic>T</italic>(<italic>e</italic><italic>i</italic>) = <italic>e</italic><italic>ϕ</italic>(<italic>i</italic>) for all <italic>i</italic> ∈ <italic>I</italic> where <italic>ϕ</italic> is conjugate to the successor function with a more sophisticated one, and to which we have not managed to find a new application yet.
</abstract>

In [Appl. Comput. Harmon. Anal., 46 (2019), 664673] 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{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.

Independence, infinite dimension, and operators

Laboratoire d’EDP, Algbre et Gomtrie spectrale

dpartement de mathmatiques, facult des sciences

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Independence, infinite dimension, and operators"}

In [Appl. Comput. Harmon. Anal., 46 (2019), 664673] O. Christensen and M. Hasannasab observed that assuming the existence of an operator T sending en to en+1...

Independence, infinite dimension, and operators

Online veröffentlicht: 01. Feb. 2023

Seitenbereich: 86 - 96

Eingereicht: 10. Aug. 2022

Akzeptiert: 21. Jan. 2023

DOI: https://doi.org/10.2478/mjpaa-2023-0006

Schlüsselwörter
vector space, operator, linear independence, dimension, ordered structures, structure

© 2023 Nizar El Idrissi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Independence, infinite dimension, and operators

Online veröffentlicht: 01. Feb. 2023

Seitenbereich: 86 - 96

Eingereicht: 10. Aug. 2022

Akzeptiert: 21. Jan. 2023

DOI: https://doi.org/10.2478/mjpaa-2023-0006

Schlüsselwörtervector space, operator, linear independence, dimension, ordered structures, structure

© 2023 Nizar El Idrissi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Schlüsselwörter
vector space, operator, linear independence, dimension, ordered structures, structure