Metric SpacesMetric space topology, as the generalization to abstract spaces of the theory of sets of points on a line or in a plane, unifies many branches of classical analysis and is necessary introduction to functional analysis. Professor Copson's book, which is based on lectures given to third-year undergraduates at the University of St Andrews, provides a more leisurely treatment of metric spaces than is found in books on functional analysis, which are usually written at graduate student level. His presentation is aimed at the applications of the theory to classical algebra and analysis; in particular, the chapter on contraction mappings shows how it provides proof of many of the existence theorems in classical analysis. |
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adherent point Banach space belongs bounded called Cauchy sequence choose closed interval closed set closure compact metric space complement complete component condition connected consider consists contains contains a point continuous mapping converges corresponding countable defined definition denoted dense depends differentiable disjoint distance E₁ element empty equation example exists extended finite finite number fixed point follows function Hence implies impossible infinite interior point intersection interval inverse least limit M₁ mean necessary non-empty subset open and closed open covering open set ordered pair positive integer positive number positive value Prove radius rational numbers real function real line real numbers relation respect result satisfies separated sequence of points sequentially compact set of points Similarly solution sphere N(a subsequence subset subspace sufficient Suppose taking tends theorem theory uniformly union unique write y₁ zero