3 Notes on Metrics , Measures , and Dimensions Stephen Semmes Department of Mathematics Rice University
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چکیده
Preface In this monograph various notions related to metric spaces are considered, including Hausdorff-type measures and dimensions, Lipschitz mappings, and the Hausdorff distance between nonempty closed and bounded subsets of a metric space. Some familiarity with basic topics in analysis such as Riemann integrals, open and closed sets, and continuous functions is assumed, as in [50, 88, 160], for instance. There is no attempt to be exhaustive in any way, and more information in a variety of directions can be found in the bibliography. ii Contents Preface ii 1 Metric spaces 1 1.
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3 Notes on Metrics , Measures , and Dimensions Stephen Semmes
Preface In this monograph various notions related to metric spaces are considered, including Hausdorff-type measures and dimensions, Lipschitz mappings, and the Hausdorff distance between nonempty closed and bounded subsets of a metric space. Some familiarity with basic topics in analysis such as Riemann integrals, open and closed sets, and continuous functions is assumed, as in [60, 100, 187],...
متن کامل3 Notes on Metrics , Measures , and Dimensions
Preface In this monograph various notions related to metric spaces are considered, including Hausdorff-type measures and dimensions, Lipschitz mappings, and the Hausdorff distance between nonempty closed and bounded subsets of a metric space. Some familiarity with basic topics in analysis such as Riemann integrals, open and closed sets, and continuous functions is assumed, as in [60, 100, 187],...
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ii Preface These notes deal with some interrelated topics involving linear algebra and geometry. The author would like to dedicate this modest monograph to Carlos " Charles " Tomei, whose influence is clearly visible. The author would also like to thank various other people for their comments and suggestions, and in particular a topics course concerning this material is taking place at Rice Uni...
متن کاملNotes on Metrics, Measures, and Dimensions
Preface In this monograph various notions of measure and content for subsets of metric spaces are considered, and both metric and topological notions of dimension. Some familiarity with basic topics in analysis such as Riemann integrals, metric spaces, and continuous functions is assumed, as in [50, 88, 160], for instance, although a number of definitions and results that are needed are reviewe...
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are also bounded functions on R for all multi-indices α, β, where d(β) denotes the degree of β, which is to say the sum of its components. Thus S(R) contains the smooth functions on R with compact support as a linear subspace. Also, S(R) contains the Gaussian functions exp(−A(x)·x), where A is a positive-definite symmetric linear transformation on R, and x · y is the usual inner product defined by
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