Let $n \geq 1$ be an integer. We denote by $\mathbb{C}[[\mathbb{Z}^n]]$ the $\mathbb{C}$-vector space of functions $f : \mathbb{Z}^n \rightarrow \mathbb{C}$. We say that $f \in \mathbb{C}[[\mathbb{Z}^n]]$ is rational if there exists a nonzero $P \in \mathbb{C}[\mathbb{Z}^n]$ such that $Pf \in \mathbb{C}[\mathbb{Z}^n]$. We say that $f$ is torsion if there exists a nonzero $P \in \mathbb{C}[\mathbb{Z}^n]$ such that $Pf = 0$. We denote by $\mathcal{R}$ the $\mathbb{C}$-vector space of rational elements and $\mathcal{T}$ the $\mathbb{C}$-vector space of torsion elements of $\mathbb{C}[[\mathbb{Z}^n]]$. In the case where $n = 1$, show that the inclusions $0 \subset \mathcal{T} \subset \mathcal{R} \subset \mathbb{C}[[\mathbb{Z}^n]]$ are strict.
Let $n \geq 1$ be an integer. We denote by $\mathbb{C}[[\mathbb{Z}^n]]$ the $\mathbb{C}$-vector space of functions $f : \mathbb{Z}^n \rightarrow \mathbb{C}$. We say that $f \in \mathbb{C}[[\mathbb{Z}^n]]$ is rational if there exists a nonzero $P \in \mathbb{C}[\mathbb{Z}^n]$ such that $Pf \in \mathbb{C}[\mathbb{Z}^n]$. We say that $f$ is torsion if there exists a nonzero $P \in \mathbb{C}[\mathbb{Z}^n]$ such that $Pf = 0$. We denote by $\mathcal{R}$ the $\mathbb{C}$-vector space of rational elements and $\mathcal{T}$ the $\mathbb{C}$-vector space of torsion elements of $\mathbb{C}[[\mathbb{Z}^n]]$.
In the case where $n = 1$, show that the inclusions $0 \subset \mathcal{T} \subset \mathcal{R} \subset \mathbb{C}[[\mathbb{Z}^n]]$ are strict.