---
tag: EBMCFQT0JN
kind: concept
name: Fréchet manifold
msc: [58B99, 46T05]
sources:
  - url: https://en.wikipedia.org/wiki/Fr%C3%A9chet_manifold
same_as:
  wikidata: Q3554820
  wikipedia: Fréchet manifold
---

## Summary

A Fréchet manifold is a space that looks locally like an open subset of a Fréchet space, with smooth changes of coordinates.
It is the natural setting for spaces of smooth maps, such as the smooth loops in a manifold or the diffeomorphisms of a compact manifold.
Calculus on Fréchet manifolds is harder than on Banach manifolds, because the inverse function theorem fails in general Fréchet spaces.
The Nash–Moser theorem gives a replacement under extra hypotheses.

## Explanations

### L1 human

Spaces of smooth functions are not Banach spaces, because no single norm controls all derivatives at once.
They are Fréchet spaces, controlled by a sequence of norms.
A Fréchet manifold is built from patches that look like pieces of such spaces.
For example, the space of all smooth loops in a sphere is a Fréchet manifold.