The Schauder Fixed-Point Theorem

An Infinite Dimensional Brouwer Theorem
  • Joel H. Shapiro
Part of the Universitext book series (UTX)


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: TheBallversion (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

Convexversion (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.


Convex Subset Normed Linear Space Closed Unit Ball Linear Topological Space Infinite Dimensional Hilbert Space 
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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Joel H. Shapiro
    • 1
  1. 1.Portland State UniversityPortlandUSA

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