Inverse function theorem in Banach spaces

Remarks

By the bounded inverse theorem, 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 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 of several-variable calculus.

Going deeper

For a student

The theorem is the main tool of local nonlinear analysis. It gives the implicit function theorem in Banach spaces, 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.