Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Independence, infinite dimension, and operators   An interesting operatorial condition ensuring the equivalence between
infinite dimension and linear independence of a sequence $(e_n)_{n \in
\mathbb{N}}$ of vectors recently appeared in a paper made by O. Christensen and
M. Hasannasab (proposition 2.3 of the paper "Frame properties of systems
arising via iterative actions of operators"). It relies on the existence of an
operator $T$ sending $e_n$ to $e_{n+1}$ for all $n \in \mathbb{N}$. In this
article, we recover this result as a particular case of a general
order-theory-based model-theoretic result. Returning to vector spaces, we then
show that, if we want to use a condition like $T(e_i)=e_{\phi(i)}$ for all $i
\in I$ where $I$ is countable as a replacement of the previous one, the
conclusion will only stay true if $\phi : I \to I$ is conjugate to the
successor function $succ : n \mapsto n+1$ defined on $\mathbb{N}$. We finally
prove a tentative generalization of the result, where we replace the condition
$T(e_i)=e_{\phi(i)}$ for all $i \in I$ where $\phi$ is conjugate to the
successor function with a more sophisticated one, and to which we have not
managed to find a new application yet.

16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
28
The categories of this paper are: alg-geom
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
19
The categories of this paper are: eess
22
The categories of this paper are: chao-dyn
26
The categories of this paper are: bayes-an
11
The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
3
The categories of this paper are: physics
17
The categories of this paper are: q-fin