Definition:Hilbert Cube

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

$\displaystyle I^\omega = \prod_{k \mathop \in \N} \left[{0 \,.\,.\, \dfrac 1 k}\right]$

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

$\displaystyle \forall x = \left\langle{x_i}\right\rangle, y = \left\langle{y_i}\right\rangle \in I^\omega: d_2 \left({x, y}\right) := \left({\sum_{k \mathop \ge 0} \left({x_k - y_k}\right)^2}\right)^{\frac 1 2}$

Also see

  • Results about the Hilbert cube can be found here.

Source of Name

This entry was named for David Hilbert.