The Contraction Mapping Theorem

A mapping f : X → X from a set X to itself has a fixed point if there is an x ∈ X such that f(x) = x. The simplest fixed point theorem is that a continuous function f : [a, b] → [a, b] has at least one fixed point. This is a consequence of the intermediate value theorem from calculus, as follows. Since f(a) ≥ a and f(b) ≤ b, we have f(b) − b ≤ 0 ≤ f(a) − a. The difference f(x)−x is continuous, so by the intermediate value theorem 0 is a value of f(x)−x for some x ∈ [a, b], and that x is a fixed point of f . (Of course, there could be more than one fixed point.) There are higher-dimensional generalizations of this result, such as the Brouwer fixed point theorem and the Lefschetz fixed point theorem. These generalizations and their proofs belong to algebraic topology. We are interested here in the most basic fixed point theorem in analysis. It is due to Banach and appeared in his Ph.D. thesis (1920, published in 1922).