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Manifolds in hilbert space? 
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#1
May808, 04:35 PM

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? 


#2
May808, 05:27 PM

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PF Gold
P: 16,092

"Locally homeomorphic to a Hilbert space" is a perfectly welldefined 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?



#3
May808, 05:38 PM

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.



#4
May1508, 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?



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