The Schauder Fixed-Point Theorem

Overview

Recall that to say a metric space has the fixed-point property means that every continuous mapping taking the space into itself must have a fixed point. In Chap. 4 we proved two versions of the Brouwer Fixed-Point Theorem: The “Ball” version (Theorem 4.1). The closed unit ball of $\mathbb{R}^{N}$ has the fixed-point property,

and the seemingly more general, but in fact equivalent

“Convex” version (Theorem 4.5). Every compact convex subset of $\mathbb{R}^{N}$ has the fixed-point property.

It turns out that the “ball” version of Brouwer’s theorem does not survive the transition to infinitely many dimensions. However all is not lost: the “convex” version does survive: compact, convex subsets of normed linear space do have the fixed-point property. This is the famous Schauder Fixed-Point Theorem (circa 1930) which will occupy us throughout this chapter. After proving the theorem we’ll use it to prove an important generalization of the Picard–Lindelöf Theorem of Chap. 3 (Theorem 3.10). The Schauder Theorem will also be important in the next chapter where it will provide a key step in the proof of Lomonosov’s famous theorem on invariant subspaces for linear operators on Banach spaces.