Hilbert Manifolds: An Infinite Dimensional Analogy to Smooth Manifolds?

[Cincinnatus]

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?

 

[Hurkyl]

"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?

 

[Cincinnatus]

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.

 

[zhentil]

Someone correct me if I'm rusty, but aren't all separable Hilbert spaces isometric to l2?

 

[mathwonk]

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.