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
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