# Hilbert cube

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## Hilbert cube

[′hil·bərt ‚kyüb]
(mathematics)
The topological space which is the cartesian product of a countable number of copies of I, the unit interval.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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In this paper we answer similar questions for when 0 is replaced with the Cantor space G or the Hilbert cube Q = [I.sup.N].
The Free Locally Convex Spaces on the Cantor Space and the Hilbert Cube
Still, for example, the triangulation of Hilbert cube manifolds--that represent the "archetypal" infinitely dimensional manifolds--is possible and, moreover, we have a by now classical Triangulation Theorem for Hilbert cube manifolds.
We can now state the Triangulation Theorem for Hilbert cube manifolds:
Theorem 3.2 (, Theorem 37.2; , [section] 6,7) Let [M.sub.Q] be a Hilbert cube manifold.
However, while in the Riemannian case the "chopping of compact pieces" criterion is based on curvature--obviously not feasible for Hilbert cube manifolds--in the case of Q-manifolds this is done via homotopy type.
Remark 3.5 While, a noted above, a general notion of fat triangulation eludes at for the moment, Theorem 3.2 allows us to formulate a practical--at least from the implementational viewpoint--definition: A triangulation of a Hilbert cube manifold MQ will be called fat if its restriction to P is fat, where P is as in Theorem 3.2 above.
That Hilbert cube manifolds do not represent a whimsical choice, but rather a natural one, is further augmented by the fact that such manifolds represent a kind of "universal space" for a large class of metric spaces:
A no less important motive to study Hilbert cube manifolds is supplied by the fact that they represent "good" representations of probability spaces, more exactly we have the following results:
Remark 4.8 Note that, if [M.sup.n] is of class [C.sup.[infinity]], then the embedding above represents--in a natural manner, via the identification [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] a submanifold of [R.sup.w] and, that, if in addition, the derivatives [[phi].sup.(k).sub.x] of the coordinate charts satisfy the relatively mild condition of being uniformly bounded, then, after a standard normalization of the said bound, of the Hilbert cube Q.
Moreover, we have shown that the finitely dimensional sampling and triangulation results also extend, albeit only in a topological sense, to more general infinite dimensional manifolds, most notably to Hilbert cube manifolds.

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