---
tag: M218MWZ62X
kind: statement
role: theorem
name: Inverse function theorem in Banach spaces
msc: [58C15, 47J07, 46G05]
uses: [06GZ956657, 06EH97V2A7, 06EKW1AKXF, 0VK6VEH345, MR5FJXR348, 0K9HSJFCMJ]
about: [06EH97V2A7, 06GZ956657]
sources:
  - url: https://en.wikipedia.org/wiki/Inverse_function_theorem
---

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, and let $X$ and $Y$ be [[Banach spaces|06GZ956657]] over $\mathbb{K}$.
Let $U \subseteq X$ be open, and let $f \colon U \to Y$ be of class $C^1$.
This means that $f$ has a [[Fréchet derivative|06EH97V2A7]] $Df(x)$ at each $x \in U$ and that $x \mapsto Df(x)$ is continuous for the [[operator norm|06EKW1AKXF]].
Let $a \in U$, and suppose that $Df(a) \colon X \to Y$ is bijective.
Then there are open sets $V \subseteq U$ with $a \in V$ and $W \subseteq Y$ with $f(a) \in W$ such that $f$ maps $V$ bijectively onto $W$.
The inverse $g \colon W \to V$ is of class $C^1$, and $Dg(f(x)) = Df(x)^{-1}$ for each $x \in V$.
If $f$ is of class $C^k$ with $k \ge 1$, then so is $g$.

## Remarks

By the [[bounded inverse theorem|MR5FJXR348]], the inverse of the bijective continuous linear map $Df(a)$ is continuous.
To solve $f(x) = y$ for $y$ near $f(a)$, one looks for a fixed point of $x \mapsto x - Df(a)^{-1}(f(x) - y)$.
Near $a$ this map is a contraction, because $Df(x)$ is close to $Df(a)$; the [[Banach fixed point theorem|0VK6VEH345]] gives existence and uniqueness.
Smoothness of the inverse follows from the formula for $Dg$ and the continuity of inversion of operators.

Example: on $X = Y = C([0,1])$ with the sup norm, let $f(u) = u + u^3$, computed pointwise.
Then $Df(0) = \id$, so each $v$ close to $0$ equals $u + u^3$ for exactly one $u$ close to $0$, and $u$ depends smoothly on $v$.

The hypothesis $C^1$ cannot be weakened to differentiability at each point.
On $\R$, the map $f(x) = x + 2x^2 \sin(1/x)$, $f(0) = 0$, has $f'(0) = 1$ but is not injective on any neighbourhood of $0$.
For $X = Y = \R^n$, the theorem is the [[inverse function theorem|0K9HSJFCMJ]] of several-variable calculus.

## Explanations

### L1 human

If the linear approximation of a smooth map at a point is invertible, the map itself is invertible near that point.
This holds even when the spaces are spaces of functions, as long as they are complete normed spaces.
So a nonlinear equation $f(x) = y$ can be solved for $y$ near $f(a)$ when its linearization at $a$ can be solved.

### L2 human

The theorem is the main tool of local nonlinear analysis.
It gives the [[implicit function theorem in Banach spaces|BTM6WJ0A52]], the local structure of Banach manifolds and the local existence for many nonlinear differential equations.
Its proof needs completeness, through the contraction principle.
In Fréchet spaces, such as $C^\infty$ function spaces, it fails in general, and the Nash–Moser theorem replaces it under extra hypotheses.