---
tag: 9K15AMGBV9
kind: concept
name: Banach manifold
msc: [58B99, 46T05]
sources:
  - url: https://en.wikipedia.org/wiki/Banach_manifold
same_as:
  wikidata: Q3554819
  wikipedia: Banach manifold
---

## Summary

A Banach manifold is a space that looks locally like an open subset of a Banach space, with smooth changes of coordinates.
It extends the notion of a smooth manifold to infinite dimension.
Spaces of maps, of curves or of operators often carry such a structure.
Calculus on Banach manifolds works much as in finite dimension, because the inverse function theorem holds in Banach spaces.

## Explanations

### L1 human

A surface is built from patches, each of which looks like a piece of the plane.
A Banach manifold is built from patches that look like pieces of a Banach space, which may be a space of functions.
The patches are glued by maps that can be differentiated.
For example, the unit sphere of the Hilbert space $\ell^2$ is a Banach manifold, and its patches are pieces of a hyperplane in $\ell^2$.