Banachsatsen

Banachsatsen, commonly known in English as Banach's fixed-point theorem or the contraction mapping principle, is a fundamental result in metric space theory and functional analysis. It gives a criterion to guarantee the existence and uniqueness of fixed points for certain self-maps and provides a constructive method to find them.

Let (X, d) be a non-empty complete metric space and let T: X → X be a contraction, meaning there exists a constant 0 ≤ q < 1 such that d(Tx, Ty) ≤ q d(x, y) for all x, y in X. Then T has a unique fixed point x* in X (i.e., Tx* = x*). Moreover, for any initial point x0 in X, the sequence x_{n+1} = T x_n converges to x*, and the convergence is geometric with rate q. A standard error bound is d(x_n, x*) ≤ (q^n / (1 - q)) d(Tx0, x0) or, equivalently, d(x_n, x_{n-1}) ≤ q^{n-1} d(x_1, x_0).

The theorem is widely used as a constructive existence result. It underpins iterative methods for solving equations, integral equations, and differential equations, and it plays a central role in numerical analysis and the theory of functional equations. In analysis, the contraction principle is employed to prove existence and uniqueness statements by reformulating problems as fixed-point problems in appropriate function spaces.

Historically, Banachsatsen is named after the Polish mathematician Stefan Banach, who established the result in the early 1920s. It remains a cornerstone of modern analysis, with numerous extensions and variations across mathematics.