---
tag: 0VK6VEH345
kind: statement
role: theorem
name: Banach fixed point theorem
aliases: [contraction-mapping-theorem, banach-fixed-point-theorem]
msc: [47H10]
uses: [0MT2S0ZX4Y, 08EWT7J0SC, 0MCK8K2D7Z]
sources:
  - url: https://en.wikipedia.org/wiki/Banach_fixed-point_theorem
    locator: Statement
same_as:
  wikidata: Q220680
  wikipedia: Banach fixed-point theorem
---

## Statement

Let $(X, d)$ be a nonempty [[complete metric space|08EWT7J0SC]] and let $f \colon X \to X$ be a [[contraction|0MT2S0ZX4Y]] with constant $k \in [0, 1)$.
Then $f$ has exactly one fixed point $x^*$, that is, a point with $f(x^*) = x^*$.
Moreover, for each $x_0 \in X$, the sequence $x_{n+1} = f(x_n)$ converges to $x^*$, and
$$d(x_n, x^*) \le \frac{k^n}{1 - k} \, d(x_1, x_0) \quad \text{for all } n \ge 0.$$

## Remarks

Proof idea: $d(x_{n+1}, x_n) \le k^n d(x_1, x_0)$, so the sequence $(x_n)$ is a [[Cauchy sequence|0MCK8K2D7Z]] by a geometric series bound. Its limit is a fixed point because $f$ is continuous. Two fixed points $x, y$ satisfy $d(x, y) \le k \, d(x, y)$, so $x = y$.
Completeness is needed: $x \mapsto x/2$ on $(0, 1]$ is a contraction without a fixed point.
The theorem gives the existence of solutions of ordinary differential equations in the Picard–Lindelöf theorem.

## History

Stefan Banach stated the theorem in 1922.

## Explanations

### L1 human

Apply the contraction again and again, starting from any point. The points get closer to each other, and in a complete space they converge.
The limit does not move under $f$, and it is the only point that does not move.