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?
"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?
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.
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.