An Introduction to Metric Spaces and Fixed Point Theory
This book provides an excellent introduction to the subject designed for readers from a variety of mathematical backgrounds. It features introductory properties of metric spaces and Banach spaces, and an appendix contains a summary of the concepts of set theory.
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assume assumption Axiom of Choice Banach lattice Banach space bounded closed convex bounded sequence Brouwer's Theorem Cauchy sequence Chapter closed balls closed convex subset compact convex subset complete metric space continuous mapping contraction mapping contradiction converges weakly convex subset countable defined Definition denote diam(D element example Exercise exists fact family of closed finite-dimensional finitely representable fixed point property fixed point set fixed point theorem hence hyperconvex metric space hyperconvex spaces implies induction integer intersection isometric Let M,d Let xn lim xn limsup metrically convex nonempty closed nonexpansive mappings nonexpansive retract norm normal structure proof of Theorem Proposition proves real numbers reflexive result satisfies Schauder basis sequence xn Show space and let space and suppose subsequence subspace super-reflexive Tn(x topological space triangle inequality ultrametric space ultranet uniformly convex weak Zorn's Lemma