Brouwer Fixed Point Theory
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.
KeywordsUnit Ball Convex Subset Close Point Normed Linear Space Point Property
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