Hilbert parallelotope

The Hilbert parallelotope Iω is a closed subset of the Hilbert spaceMathworldPlanetmath 2 (The symbol ’’ has been prefixed to indicate that the field of scalars is .) defined as


As a topological spaceMathworldPlanetmath, Iω is homeomorphic to the productPlanetmathPlanetmathPlanetmath of a countably infiniteMathworldPlanetmath number of copies of the closed intervalMathworldPlanetmath [0,1]. By TychonoffPlanetmathPlanetmath’s theoremMathworldPlanetmath, this product is compactPlanetmathPlanetmath, so the Hilbert parallelotope is a compact subset of Hilbert space. This fact also explains the notation Iω.

The Hilbert parallelotope enjoys a remarkable universality property — every second countable metric space is homeomorphic to a subset of the Hilbert parallelotope. Since second countability is hereditary, the converseMathworldPlanetmath is also true — every subset of the Hilbert parallelotope is a second countable metric space.

Title Hilbert parallelotope
Canonical name HilbertParallelotope
Date of creation 2013-03-22 14:38:32
Last modified on 2013-03-22 14:38:32
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Definition
Classification msc 46C05
Synonym Hilbert cube