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| Preface | |
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| Metric Spaces and Normed Linear Spaces | |
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| Definitions and Examples | |
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| Metric spaces, normed linear spaces | |
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| Metrics generated by a norm | |
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| Co-ordinate, sequence and function spaces | |
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| Semi-normed linear spaces | |
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| Exercises | |
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| Balls and Boundedness | |
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| Balls and spheres in metric spaces and normed linear spaces, relating norms and balls | |
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| Boundedness, diameter | |
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| Distances between sets | |
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| Exercises | |
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| Limit Processes | |
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| Convergence and Completeness | |
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| Convergence of sequences, characterisation in finite dimensional normed linear spaces, uniform convergence | |
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| Equivalent metrics and norms | |
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| Cauchy sequences, completeness | |
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| Convergence of series | |
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| Exercises | |
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| Cluster Points and Closure | |
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| Cluster points, closed sets | |
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| Relating closed to complete | |
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| Closure, density, separability | |
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| The boundary of a set | |
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| Exercises | |
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| Application: Banach's Fixed Point Theorem | |
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| Fixed points, Banach's Fixed Point Theorem | |
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| Application in real analysis | |
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| Application in linear algebra | |
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| Application in the theory of differential equations | |
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| Picard's Theorem | |
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| Application in the theory of integral equations | |
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| Fredholm integral equations, Volterra integral equations | |
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| Exercises | |
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| Continuity | |
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| Continuity in Metric Spaces | |
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| Local continuity, characterisation of continuity by sequences, algebra of continuous mappings | |
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| Global continuity characterised by inverse images | |
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| Isometrics, homeomorphisms | |
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| Uniform continuity | |
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| Exercises | |
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| Continuous Linear Mappings | |
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| Characterisation of continuity of linear mappings, linear mappings on finite dimensional normed linear spaces, continuity of linear functionals | |
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| Topological isomorphisms, isometric isomorphisms | |
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| Exercises | |
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| Compactness | |
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| Sequential Compactness in Metric Spaces | |
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| Properties of compact sets | |
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| Characterisation in finite dimensional normed linear spaces, Riesz Theorem | |
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| Application in approximation theory | |
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| Alternative forms of compactness, total boundedness, ball cover compactness | |
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| Separability | |
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| Exercises | |
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| Continuous Functions on Compact Metric Spaces | |
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| Heine's Theorem, Dini's Theorem | |
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| The structure of the real Banach space (C [a, b], [double vertical bar][middle dot][double vertical bar][subscript infinity] | |
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| The Weierstrass Approximation Theorem | |
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| The structure of the Banach space (C(X), [double vertical bar][middle dot][double vertical bar][subscript infinity] where (X, d) is a compact metric space | |
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| Compactness in (C(X), [double vertical bar][middle dot][double vertical bar][subscript infinity] | |
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| Equicontinuity, The Ascoli-Arzela Theorem, Peano's Theorem | |
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| Exercises | |
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| The Metric Topology | |
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| The Topological Analysis of Metric Spaces | |
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| Open sets and their properties, base for a topology | |
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| Equivalent metrics | |
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| Relation to closed sets | |
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| The interior of a set | |
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| The characterisation of continuous mappings by inverse images | |
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| Topological compactness | |
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| Separability, the normal topological structure | |
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| Exercises | |
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| Appendices | |
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| The real analysis background | |
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| The set theory background | |
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| The linear algebra background | |
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| Index to Notation | |
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| Index | |