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Difference between revisions of "Hilbert cube"

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The subspace of the [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047240/h0472401.png" /> consisting of all the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047240/h0472402.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047240/h0472403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047240/h0472404.png" />. The Hilbert cube is a [[Compactum|compactum]] and is topologically equivalent (homeomorphic) to the Tikhonov product of a countable system of intervals, i.e. to the [[Tikhonov cube|Tikhonov cube]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047240/h0472405.png" />. It is a [[Universal space|universal space]] in the class of metric spaces with a countable base (Urysohn's metrization theorem).
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The subspace of the [[Hilbert space|Hilbert space]] $l_2$ consisting of all the points $x=(x_1,x_2,\ldots)$ for which $0\leq x_n\leq(1/2)^n$, $n=1,2,\ldots$. The Hilbert cube is a [[Compactum|compactum]] and is topologically equivalent (homeomorphic) to the Tikhonov product of a countable system of intervals, i.e. to the [[Tikhonov cube|Tikhonov cube]] $I^{\aleph_0}$. It is a [[Universal space|universal space]] in the class of metric spaces with a countable base (Urysohn's metrization theorem).
  
  

Latest revision as of 21:09, 12 April 2014

The subspace of the Hilbert space $l_2$ consisting of all the points $x=(x_1,x_2,\ldots)$ for which $0\leq x_n\leq(1/2)^n$, $n=1,2,\ldots$. The Hilbert cube is a compactum and is topologically equivalent (homeomorphic) to the Tikhonov product of a countable system of intervals, i.e. to the Tikhonov cube $I^{\aleph_0}$. It is a universal space in the class of metric spaces with a countable base (Urysohn's metrization theorem).


Comments

The topology of the Hilbert cube is studied in the field of infinite-dimensional topology (cf. Infinite-dimensional space). This is a rich and fruitful area of investigation.

See [a1] for an excellent introduction and references.

References

[a1] J. van Mill, "Topology; with an introduction to infinite-dimensional spaces" , North-Holland (1988)
How to Cite This Entry:
Hilbert cube. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hilbert_cube&oldid=12075
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article