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Manifolds in hilbert space?

by Cincinnatus
Tags: hilbert, manifolds, space
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May8-08, 04:35 PM
Cincinnatus's Avatar
P: 395
Can you define a space that is locally homeomorphic to an infinite dimensional Hilbert space analogously to how you define a (smooth) manifold by an atlas defining local homeomorphisms to R^n?

So the charts would just map to the Hilbert space rather than R^n. Could the rest of the definition be left the same?
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May8-08, 05:27 PM
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Hurkyl's Avatar
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"Locally homeomorphic to a Hilbert space" is a perfectly well-defined condition. Such objects might not be interesting to study -- e.g. it completely ignores the algebraic structure of Hilbert space. (compare: in the definition of a manifold, R^n is used for its simple geometry, and not for its vector space structure) What is the the intended application?
May8-08, 05:38 PM
Cincinnatus's Avatar
P: 395
In machine learning, people are interested in taking sets of documents, images or whatever and defining a similarity measure on them (kernel) which gives rise to a metric in the Hilbert space represented by that kernel. I was wondering if the space you get in this way couldn't have more complicated geometric structure.

May15-08, 04:48 AM
P: 491
Manifolds in hilbert space?

Someone correct me if I'm rusty, but aren't all separable Hilbert spaces isometric to l2?
May15-08, 01:33 PM
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mathwonk's Avatar
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the theory of hilbert manifolds and infinite dimensional morse theory is mroe than 40 years old. see richard palais, morse theory on hilbert manifolds, topology 1963, and a 1964 article in BAMS by steven smale.

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