Definition ($\Gamma$)

For two vector spaces $\mathbb{X},\mathbb{Y}$ let $\Gamma(X;Y)$ denote the set of all maps $\gamma:\mathbb{X}\to \mathbb{Y}$ such that $\forall x\in \mathbb{X}$, $\gamma(x)\in \mathbb{Y}$.

Definition ($l_{p}$)

$$l_{p}(\mathbb{Z}_{+};\mathbb{R}):=\left\{ x\in\Gamma(\mathbb{Z}_{+};\mathbb{R}):\lVert x \rVert _{p}=\left( \sum_{i\in \mathbb{Z}_{+}}\left| x(i) \right| ^{p} \right)^{1/p}<\infty \right\}$$