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

(x_n)

that satisfies

0 ≤ x_n ≤ 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.