I'm struggling with the notation for space $\mathcal{X}$ of N-tuples of elements of $\ell^2(\mathbb{Z})$, where $\ell^2(\mathbb{Z})$ is the space of sequences of finite energy. As an example, the following $\mathbb{x}[n]$ is an element of the space I'm looking for, when $N=2$:
$$\mathbb{x} \in \mathcal{X}, \mathbb{x}[n]:=(x_1[n], x_2[n])$$
where $x_1,_2 \in\ell^2(\mathbb{Z})$. Essentially, for each $n$, $\mathbb{x}[n]$ is a tuple of $N$ elements, given by $(x_1[n], x_2[n], \ldots, x_N[n])$.
I'm tempted to write something like $\mathcal{X}=[\ell^2(\mathbb{Z})]^N$ or $\mathcal{X}=\mathcal{C}^{(N\times\mathbb{Z})}$ but it all looks strange.