Hilbert Cube - The Hilbert Cube As A Metric Space

The Hilbert Cube As A Metric Space

It's sometimes convenient to think of the Hilbert cube as a metric space, indeed as a specific subset of a separable Hilbert space (i.e. a Hilbert space with a countably infinite Hilbert basis). For these purposes, it is best not to think of it as a product of copies of, but instead as

× × × ···;

as stated above, for topological properties, this makes no difference. That is, an element of the Hilbert cube is an infinite sequence

(xn)

that satisfies

0 ≤ xn ≤ 1/n.

Any such sequence belongs to the Hilbert space ℓ2, so the Hilbert cube inherits a metric from there. One can show that the topology induced by the metric is the same as the product topology in the above definition.

Read more about this topic:  Hilbert Cube

Famous quotes containing the word space:

    The woman’s world ... is shown as a series of limited spaces, with the woman struggling to get free of them. The struggle is what the film is about; what is struggled against is the limited space itself. Consequently, to make its point, the film has to deny itself and suggest it was the struggle that was wrong, not the space.
    Jeanine Basinger (b. 1936)