Hilbert manifold
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In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to a Hilbert space. Hilbert manifolds are one possibility of extending manifolds to infinite dimensions.
For a formal definition of a Hilbert manifold, see the article on Banach manifolds and replace "Banach space" with "Hilbert space" throughout.
[edit] Examples
- Any Hilbert space H is a Hilbert manifold with a single global chart given by the identity function on H. Moreover, since H is a vector space, the tangent space TpH to H at any point p ∈ H is canonically isomorphic to H itself, and so has a natural inner product, the "same" as the one on H. Thus, H can be given the structure of a Riemannian manifold with metric
- where
denotes the inner product in H.
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