Hilbert cube

The Hilbert cube is a named after the mathematician David Hilbert topological space that generalizes the known from the space of intuition dice on an infinite number of dimensions.

Definition

The Hilbert cube is the product space equipped with the product topology, that is, more precisely:

  • Is the set of all sequences with for all.
  • A sequence in, with, converges to one against, if for all indices.

Properties

  • The Hilbert cube is connected and path-connected, since these properties are transmitted to product spaces.
  • The Hilbert cube is a compact Hausdorff space, as follows immediately from the theorem of Tychonoff.
  • The Hilbert cube is metrizable topology defining a metric is by
  • Like all compact metrizable spaces is the Hilbert cube is separable and satisfies the Abzählbarkeitsaxiomen. The amount
  • The lebesgue'sche covering dimension of the Hilbert cube is infinite, since for each of the Hilbert cube contains the homeomorphic to the subspace, therefore, a dimension must have for all and that is.

Universal property

Compact spaces with a countable base

The Hilbert cube is according to the above properties a compact Hausdorff space with a countable base. is universal with respect to these properties in the sense that it contains a copy of each such space. The following applies:

  • Every compact Hausdorff space with a countable base is homeomorphic to a closed subspace of the Hilbert cube.

Polish spaces

Also Polish spaces can be embedded in the Hilbert cube. The following applies:

  • The Polish spaces up to homeomorphism precisely the amounts in the Hilbert cube.
  • The compact Polish spaces up to homeomorphism precisely the closed sets in the Hilbert cube.

The Hilbert cube in l2

A homeomorphic copy of the Hilbert cube is found in the Hilbert space of quadratsummierbaren consequences. define

Then is a homeomorphism if you know with the subspace topology of the norm topology of the Hilbert space. Note that no zero environment is in, because no standard ball. Furthermore, coincide on the relative norm topology and the relative weak topology.

Alternative definitions of the Hilbert cube or would or provided with the product topology. Such a definition would be even a subset of the Hilbert space. The first variant is used, there speaks the author because of the different side lengths not from the Hilbert cube but from the Hilbert cube, as in where the third variant is used for the definition.

Euclidean space Sequence space
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