\begin{definition}[Definition:Hilbert Sequence Space]
Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} x_i^2$ is convergent.
Let $d_2: A \times A: \to \R$ be the real-valued function defined as:
:$\ds \forall x = \sequence {x_i}, y = \sequence {y_i} \in A: \map {d_2} {x, y} := \paren {\sum_{k \mathop \ge 0} \paren {x_k - y_k}^2}^{\frac 1 2}$
The metric space $\struct {A, d_2}$ is the '''Hilbert sequence space on $\R$''' and is denoted $\ell^2$.
\end{definition}