Modern Analysis and Topology

Häftad, Engelska, 1995

534 kr

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Beskrivning

The purpose of this book is to provide an integrated development of modern analysis and topology through the integrating vehicle of uniform spaces. The reader should have taken an advanced calculus course and an introductory topology course. It is intended that a subset of the book could be used for an upper-level undergraduate course whereas much of the full text would be suitable for a one-year graduate class. An attempt has been made to document the history of all the central ideas and references and historical notes are embedded in the text. These can lead the interested reader to the foundational sources where these ideas emerged.

Produktinformation

Utgivningsdatum:1995-06-23
Mått:155 x 235 x 24 mm
Vikt:663 g
Format:Häftad
Språk:Engelska
Serie:Universitext
Antal sidor:444
Upplaga:1995
Förlag:Springer-Verlag New York Inc.
ISBN:9780387979861

Utforska kategorier

Innehållsförteckning

1: Metric Spaces.- 1.1 Metric and Pseudo-Metric Spaces.- 1.2 Stone’s Theorem.- 1.3 The Metrization Problem.- 1.4 Topology of Metric Spaces.- 1.5 Uniform Continuity and Uniform Convergence.- 1.6 Completeness.- 1.7 Completions.- 2: Uniformities.- 2.1 Covering Uniformities.- 2.2 Uniform Continuity.- 2.3 Uniformizability and Complete Regularity.- 2.4 Normal Coverings.- 3: Transfinite Sequences.- 3.1 Background.- 3.2 Transfinite Sequences in Uniform Spaces.- 3.3 Transfinite Sequences and Topologies.- 4: Completeness, Cofinal Completeness And Uniform Paracompactness.- 4.1 Introduction.- 4.2 Nets.- 4.3 Completeness, Cofinal Completeness and Uniform Paracompactness.- 4.4 The Completion of a Uniform Space.- 4.5 The Cofinal Completion or Uniform Paracompactification.- 5: Fundamental Constructions.- 5.1 Introduction.- 5.2 Limit Uniformities.- 5.3 Subspaces, Sums, Products and Quotients.- 5.4 Hyperspaces.- 5.5 Inverse Limits and Spectra.- 5.6 The Locally Fine Coreflection.- 5.7 Categories and Functors.- 6: Paracompactifications.- 6.1 Introduction.- 6.2 Compactifications.- 6.3 Tamano’s Completeness Theorem.- 6.4 Points at Infinity and Tamano’s Theorem.- 6.5 Paracompactifications.- 6.6 The Spectrum of ?X.- 6.7 The Tamano-Morita Paracompactification.- 7: Realcompactifications.- 7.1 Introduction.- 7.2 Realcompact Spaces.- 7.3 Realcompactifications.- 7.4 Realcompact Spaces and Lindelöf Spaces.- 7.5 Shirota’s Theorem.- 8: Measure And Integration.- 8.1 Introduction.- 8.2 Measure Rings and Algebras.- 8.3 Properties of Measures.- 8.4 Outer Measures.- 8.5 Measurable Functions.- 8.6 The Lebesgue Integral.- 8.7 Negligible Sets.- 8.8 Linear Functional and Integrals.- 9: Haar Measure In Uniform Spaces.- 9.1 Introduction.- 9.2 Haar Integrals and Measures.- 9.3 Topological Groups andUniqueness of Haar Measures.- 10: Uniform Measures.- 10.1 Introduction.- 10.2 Prerings and Loomis Contents.- 10.3 The Haar Functions.- 10.4 Invariance and Uniqueness of Loomis Contents and Haar Measures.- 10.5 Local Compactness and Uniform Measures.- 11: Spaces Of Functions.- 11.1 LP -spaces.- 11.2 The Space L2(?) and Hilbert Spaces.- 11.3 The Space LP(?) and Banach Spaces.- 11.4 Uniform Function Spaces.- 12: Uniform Differentiation.- 12.1 Complex Measures.- 12.2 The Radon-Nikodym Derivative.- 12.3 Decompositions of Measures and Complex Integration.- 12.4 The Riesz Representation Theorem.- 12.5 Uniform Derivatives of Measures.