Definition:Hilbert Cube

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Definition 1

The Hilbert cube $\struct {I^\omega, d_2}$ is the subspace of the Hilbert sequence space $I^\omega$ defined as:

$\ds I^\omega = \prod_{k \mathop \in \N_{>0} } \closedint 0 {\dfrac 1 k}$

under the same metric as that of the Hilbert sequence space:

$\ds \forall x = \sequence {x_i}, y = \sequence {y_i} \in I^\omega: \map {d_2} {x, y} := \paren {\sum_{k \mathop \in \N_{>0} } \paren {x_k - y_k}^2}^{\frac 1 2}$

Definition 2

The Hilbert cube, denoted by $I^\omega$, is defined as:

$\ds I^\omega := \set {\sequence {x_n}_{n \mathop \in \N_{> 0} } \in \R^\N: 0 \le x_n \le \frac 1 n}$

Also denoted as

The Hilbert cube is also seen denoted as $C$ or $Q$.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ $I^\omega$ is preferred as it is descriptive and fairly unambiguous.

Also see

  • Results about the Hilbert cube can be found here.

Source of Name

This entry was named for David Hilbert.