Like the question asks, if I have a subspace, F, of $ \mathbb{R}^\mathbb{N}$ (the set of all infinite sequences that follow $(a_1, a_2,...)$) such that the infinite sequence must satisfy $$a_i + a_{i+1} = a_{i+2} \text{ for all } i \geq 2$$ ow could I find a basis for F?
My initial thought process stemmed from the idea that the basis must be a linearly independent set of sequences such that multiples of one sequence cannot produces multiples of the other sequences in the basis, but there's no specified starting point of the sequence satisfying F, and there's no specification that the sequence must consist of real numbers.
My assumption is that the basis would be something like $\{(a_1 , a_2,...): a_1 \in \mathbb{R} \text{ and } a_2 \in \mathbb{R}\}$ where $a_1 \text{ and } a_2$ must be linearly independent, but I don't necessarily know how to describe that in mathematical terms for the basis.