Definition:Hilbert Sequence Space

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This page is about the metric space formed of real sequences under the Euclidean metric. For other uses, see Definition:Hilbert 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

  • Results about the Hilbert sequence space can be found here.


Source of Name

This entry was named for David Hilbert.


Sources