# An Introduction to Metric Spaces and Fixed Point Theory by Mohamed A. Khamsi By Mohamed A. Khamsi

Offers up to date Banach house results.
* beneficial properties an in depth bibliography for outdoor reading.
* presents distinctive workouts that elucidate extra introductory fabric.

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C) => (E). Proof. Assume (E) is false. Then Va: € M 3 g(x) 6 M such that x < g(x). It follows that d{x,g{x)) < ψ{χ) - (E) uses the Axiom of Choice, whereas the proof that (E) => (C) does not. In fact, these two theorems are equivalent only if one (*) 56 CHAPTER 3. METRIC CONTRACTION PRINCIPLES assumes (as we do) the Axiom of Choice.

METRIC SPACES Clearly 0 < d'(x,y) < 1. Also d'(x,y) = 0 & x — y, and d'(x,y) - d'(y,x). There are two ways to see that the triangle inequality holds. One way is the direct computation: d'(x,z) =d(x,z)/[l+d(x,z)] = 1 - 1 / ( 1 + d(x,z)] 0, f'(t) = — l — ^ > 0 and f"(t) = j- -2 rz < 0.

In particular, some member U of ÏÀ2n lies in U(c; 1/n); hence KnW < M2„(C) < M2n(^2n), contradicting the definition of μ^Λ^η)- ■ It is possible to give a much quicker proof of the above theorem using Zorn's Lemma. ) The proof just given is Menger's original and it predates the discovery of Zorn's Lemma. In fact, with Zorn's Lemma it is quite easy to prove the following. 18 Let M be a metrically convex metric space, and suppose the intersection of every descending chain of closed metrically convex subsets of M is itself metrically convex.