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 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.
KeywordsUnit Ball Convex Subset Fixed Point Theorem Close Point Normed Linear Space
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