---
tag: R8DMMW0E9M
kind: concept
name: Hilbert manifold
msc: [58B99, 46T05, 57N20]
specializes: [9K15AMGBV9]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert_manifold
same_as:
  wikidata: Q3554824
  wikipedia: Hilbert manifold
---

## Summary

A Hilbert manifold is a [[Banach manifold|9K15AMGBV9]] whose charts take values in [[Hilbert spaces|HM2HHE3J51]].
Inner products give partitions of unity, orthogonal complements and Riemannian metrics, so much of finite-dimensional differential topology carries over.
Spaces of paths and loops of Sobolev class $H^1$ in a finite-dimensional manifold are Hilbert manifolds.
This makes them the natural setting for Morse theory in infinite dimensions.
Take an infinite-dimensional separable model and a separable metrizable manifold.
Then the [[open embedding theorem|JWCDDMJE3K]] shows that it is an open subset of its model space, up to diffeomorphism.

## Explanations

### L1 human

A Hilbert manifold is built from patches that look like pieces of a Hilbert space, glued by smooth maps.
A Hilbert space is a possibly infinite-dimensional space in which lengths and angles make sense, such as the space $\ell^2$ of square-summable sequences.
The unit sphere of $\ell^2$, the set of sequences with $\sum x_n^2 = 1$, is a Hilbert manifold.
Near each of its points it looks like the hyperplane orthogonal to that point.