# Brouwer Fixed Point Theory

## Abstract

A topological space *Y* has the *fixed point property*, abbreviated *fpp*, if every map (continuous function) *f : Y → Y* has a fixed point, that is, *f(y) = y* for some *y ∈ Y*. The fixed point property is a topological property in the sense that it is preserved by homeomorphisms. That is, it’s easy to see that if a space *Y* has the *fpp* and *Z* is homeomorphic to *Y*, then *Z* also has the *fpp*. The Schauder fixed point theorem, quoted in Chapter 1 as the key to the topological proof of the Cauchy-Peano theorem, states that a compact, convex subset of a normed linear space has the *fpp*. We’ll prove the Schauder theorem, along with a very useful extension of it, in Chapter 4. The proof is accomplished in two steps: first prove a finite-dimensional version of Schauder’s theorem, then generalize to normed linear spaces in general. This chapter will be devoted to the first of these steps.

## Keywords

Unit Ball Convex Subset Fixed Point Theorem Close Point Normed Linear Space## Preview

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