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 e_n+1 for all n ∈ ℕ (where (e_n)_n∈ℕ is a sequence of vectors) guarantees that (e_n)_n∈ℕ is linearly independent if and only if dim{e_n}_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(e_i) = e_ϕ(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(e_i) = e_ϕ(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.