Definition:Hilbert Sequence Space

Definition
Let $A$ be the set of all real sequences $\left\langle{x_i}\right\rangle$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent.

Let $d_2: A \times A: \to \R$ be the real-valued function defined as:
 * $\displaystyle \forall x = \left\langle{x_i}\right\rangle, y = \left\langle{y_i}\right\rangle \in A: d_2 \left({x, y}\right) := \left({\sum_{k \mathop \ge 0} \left({x_k - y_k}\right)^2}\right)^{\frac 1 2}$

The metric space $\left({A, d_2}\right)$ is the Hilbert sequence space on $\R$ and is denoted $\ell^2$.

Also see

 * Definition:Sequence Space


 * Hilbert Sequence Space is Metric Space