Definition:Hilbert Sequence Space

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This page is about Hilbert Sequence Space. For other uses, see Hilbert 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 $\ell^2$.

Also see

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

Source of Name

This entry was named for David Hilbert.