Show that $V$ is a vector subspace of $\mathbf{C}^{\mathbf{Z}}$. Given $f \in \mathbf{C}^{\mathbf{Z}}$, we define $E(f) \in \mathbf{C}^{\mathbf{Z}}$ by $E(f)(k) = f(k+1), k \in \mathbf{Z}$.
Show that $V$ is a vector subspace of $\mathbf{C}^{\mathbf{Z}}$. Given $f \in \mathbf{C}^{\mathbf{Z}}$, we define $E(f) \in \mathbf{C}^{\mathbf{Z}}$ by $E(f)(k) = f(k+1), k \in \mathbf{Z}$.