Definition:Generalized Hilbert Sequence Space

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This page is about Generalized Hilbert Sequence Space. For other uses, see Hilbert Sequence Space.

Definition

Let $\alpha$ be an infinite cardinal.

Let $I$ be an indexed set of cardinality $\alpha$.


Let $A$ be the set of all real-valued functions $x : I \to \R$ such that:

$(1)\quad \set{i \in I: x_i \ne 0}$ is countable
$(2)\quad$ the generalized sum $\ds \sum_{i \mathop \in I} x_i^2$ is a convergent net.

Let $d_2: A \times A \to \R$ be the real-valued function defined as:

$\ds \forall x = \family {x_i}, y = \family {y_i} \in A: \map {d_2} {x, y} := \paren {\sum_{i \mathop \in I} \paren {x_i- y_i}^2}^{\frac 1 2}$


The metric space $\struct {A, d_2}$ is the generalized Hilbert sequence space on $\R$ of weight $\alpha$ and is denoted $H^\alpha$.


Also see

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


Source of Name

This entry was named for David Hilbert.


Sources

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