Let $A = \sum_{k=0}^{p} a_k X^k$ be an element of $\mathbb{K}[X]$, of degree $p \geqslant 0$, which without loss of generality we assume to be monic.
Prove that $\mathcal{R}_A(\mathbb{K})$ is a vector subspace of dimension $p$ of $\mathbb{K}^{\mathbb{N}}$ and that it is stable under $\sigma$ (we do not ask here to determine a basis of $\mathcal{R}_A(\mathbb{K})$, as this is the object of the following questions).