# 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 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 that theorem and more in Chapter 4. The proof is accomplished in two steps: first prove a finite-dimensional fixed point theorem, then generalize to normed linear spaces. This chapter will be devoted to the first of these steps.

## Keywords

Unit Ball Convex Subset Close Point Normed Linear Space Point Property## Preview

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