# Category:Hilbert Cube

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This category contains results about **the Hilbert cube**.

Definitions specific to this category can be found in **Definitions/Hilbert Cube**.

### 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:

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

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Hilbert Cube"

The following 8 pages are in this category, out of 8 total.