Groupoid Metrization Theory: With Applications to Analysis on Quasi-Metric Spaces and Functional Analysis (Applied and Numerical Harmonic Analysis)

!Cover 1 Groupoid Metrization Theory 4 ANHA Series Preface 6 Preface 10 Contents 12 Chapter 1 Introduction 14 1.1 Overview 15 1.2 First Look at the Groupoid Metrization Theorem 19 Chapter 2 Semigroupoids and Groupoids 24 2.1 Algebraic Considerations 24 2.1.1 Semigroupoids 25 2.1.2 Groupoids 37 2.2 Topological Considerations 58 Chapter 3 Quantitative Metrization Theory 73 3.1 Regularizing Quasisubadditive Functions 73 3.2 Main Groupoid Metrization Theorem 103 3.2.1 Formulation of Groupoid Metrization Theorem 103 3.2.2 Proof of GMT 110 3.2.3 More on the Relationship Between GMT and Macías–Segovia, Aoki–Rolewicz, and Alexandroff–Urysohn Theorems 120 3.2.4 Connections with Homogeneous Groups 125 3.3 Metrization Theory in Semigroupoid Setting 128 3.3.1 A Sharp Semigroupoid Metrization Theorem 128 3.3.2 An Application to Analytic Capacity 136 3.3.3 Metrization Results with Additional Constraints 139 3.4 A Sharp Metrization Result for Quasimetric Spaces 156 Chapter 4 Applications to Analysis on Quasimetric Spaces 164 4.1 Category of Quasimetric Spaces 164 4.2 Extensions of Hölder Functions 167 4.3 Separation Properties of Hölder Functions 175 4.4 Density and Embedding Properties of Hölder Functions 177 4.5 Regularized Distance Function to a Set 185 4.6 Whitney-Like Partitions of Unity via Hölder Functions 189 4.7 Smoothness Indexes of a Quasimetric Space 207 4.8 Distribution Theory on Quasimetric Spaces 257 4.9 Hardy Spaces on Ahlfors-Regular Quasimetric Spaces 261 4.10 Approximation to the Identity on Ahlfors-Regular Quasimetric Spaces 272 4.11 Bi-Lipschitz Euclidean Embeddings of Quasimetric Spaces 281 4.12 Quasimetric Version of Kuratowski's and Fréchet's Embedding Theorems 288 4.13 Pompeiu–Hausdorff Quasidistance on Quasimetric Spaces 290 4.14 Gromov–Pompeiu–Hausdorff Distance Between Quasimetric Spaces 294 Chapter 5 Nonlocally Convex Functional Analysis 305 5.1 Formulation of Results 305 5.2 Examples 310 5.3 Abstract Capacitary Estimates 321 5.4 Abstract Completeness Results 327 5.4.1 Linear Background 327 5.4.2 Boolean Algebra Background 342 5.5 Absolute Continuity of a Measure with Respect to a Capacity 358 5.6 Embeddings and Pointwise Convergence 363 5.7 Separability 365 Chapter 6 Functional Analysis on Quasi-Pseudonormed Groups 369 6.1 Topological and Algebraical Preliminaries 370 6.2 Quasi-Pseudonormed Groups and an Extension of the Birkhoff–Kakutani Theorem 387 6.3 Quotient, Pullback, and Push-Forward Quasi-Pseudonorms 400 6.4 A Quantitative Open Mapping Theorem 416 6.5 Closed Graph Theorem 440 6.6 Uniform Boundedness Principle 452 6.7 A Unified Approach to OMT/CGT/UBP 457 6.8 Further Applications 466 064_007 470 References 470 Symbol Index 475 Subject Index 478 Theorem Index 482 Author Index 484 The topics in this research monograph are at the interface of several areas of mathematics such as harmonic analysis, functional analysis, analysis on spaces of homogeneous type, topology, and quasi-metric geometry. The presentation is self-contained with complete, detailed proofs, and a large number of examples and counterexamples are provided. Unique features of Metrization Theory for Groupoids: With Applications to Analysis on Quasi-Metric Spaces and Functional Analysis include: * treatment of metrization from a wide, interdisciplinary perspective, with accompanying applications ranging across diverse fields; * coverage of topics applicable to a variety of scientific areas within pure mathematics; * useful techniques and extensive reference material; * includes sharp results in the field of metrization. Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties. * coverage of topics applicable to a variety of scientific areas within pure mathematics; * useful techniques and extensive reference material; * includes sharp results in the field of metrization. Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties. * useful techniques and extensive reference material; * includes sharp results in the field of metrization. Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties. * includes sharp results in the field of metrization. Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties. Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties.