Metric spaces : a companion to analysis / / Robert Magnus

• Record Nr.: UNISA996466417703316
• Autore: Magnus Robert
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer Nature Switzerland AG, , [2022]; ©2022
• ISBN: 9783030949464; 9783030949457
• Descrizione fisica: 1 online resource (258 pages)
• Collana: Springer undergraduate mathematics series
• Disciplina: 514.32
• Soggetti: Metric spaces; Mathematics; Espais mètrics; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Includes index.
• Nota di contenuto: Intro -- Preface -- Contents -- Preliminaries on Sets -- Basic Relations -- Basic Operations -- Writing Predicates -- Set-Building Rules -- Relations and Functions -- Cardinals -- Other Notions -- 1 Metric Spaces -- 1.1 Metrics -- 1.1.1 Rationale for Metrics -- 1.1.2 Defining Metric Space -- 1.1.3 Exercises -- 1.2 Examples of Metric Spaces -- 1.2.1 Normed Spaces -- 1.2.2 Subspaces -- 1.2.3 Examples -- Not Subspaces of Normed Spaces -- 1.2.4 Pseudometrics -- 1.2.5 Cauchy-Schwarz, Hölder, Minkowski -- 1.2.6 Exercises -- 1.3 Cantor's Middle Thirds Set -- 1.3.1 Exercises -- 1.4 The Normed Spaces of Functional Analysis -- 1.4.1 Sequence Spaces -- 1.4.2 Function Spaces -- 1.4.3 Spaces of Continuous Functions -- 1.4.4 Spaces of Integrable Functions -- 1.4.5 Hölder's and Minkowski's Inequalities for Integrals -- 1.4.6 Exercises -- 2 Basic Theory of Metric Spaces -- 2.1 Balls in a Metric Space -- 2.1.1 Limit of a Convergent Sequence -- 2.1.2 Uniqueness of the Limit -- 2.1.3 Neighbourhoods -- 2.1.4 Bounded Sets -- 2.1.5 Completeness -- a Key Concept -- 2.1.6 Exercises -- 2.2 Open Sets, and Closed -- 2.2.1 Open Sets -- 2.2.2 Union and Intersection of Open Sets -- 2.2.3 Closed Sets -- 2.2.4 Union and Intersection of Closed Sets -- 2.2.5 Characterisation of Open and Closed Sets by Sequences ; -- 2.2.6 Interior, Closure and Boundary -- 2.2.7 Limit Points of Sets -- 2.2.8 Characterisation of Closure by Limit Points -- 2.2.9 Subspaces -- 2.2.10 Open and Closed Sets in a Subspace -- 2.2.11 Exercises -- 2.3 Continuous Mappings -- 2.3.1 Defining Continuity -- 2.3.2 New Views of Continuity -- 2.3.3 Limits of Functions -- 2.3.4 Characterising Continuity by Sequences -- 2.3.5 Lipschitz Mappings -- 2.3.6 Examples of Continuous Functions -- 2.3.7 Exercises -- 2.4 Continuity of Linear Mappings -- 2.4.1 Continuity Criterion -- 2.4.2 Operator Norms -- 2.4.3 Exercises.; 2.5 Homeomorphisms and Topological Properties -- 2.5.1 Equivalent Metrics -- 2.5.2 Exercises -- 2.6 Topologies and σ-Algebras -- 2.6.1 Order Topologies -- 2.6.2 Exercises -- 2.6.3 Pointers to Further Study -- 2.7 () Mazur-Ulam -- 2.7.1 Exercises -- 3 Completeness of the Classical Spaces -- 3.1 Coordinate Spaces and Normed Sequence Spaces -- 3.1.1 Completeness of Rn -- 3.1.2 Completeness of p -- 3.1.3 Exercises -- 3.2 Product Spaces -- 3.2.1 Finitely Many Factors -- 3.2.2 Infinitely Many Factors -- 3.2.3 The Space 2N+ and the Cantor Set -- 3.2.4 Subspaces of Complete Spaces -- 3.2.5 Exercises -- 3.3 Spaces of Continuous Functions -- 3.3.1 Uniform Convergence -- 3.3.2 Series in Normed Spaces -- 3.3.3 The Weierstrass M-Test -- 3.3.4 The Spaces C(R) and Cp(R) -- 3.3.5 Exercises -- 3.4 () Rearrangements -- 3.4.1 Vector Series -- 3.4.2 Exercises -- 3.4.3 Pointers to Further Study -- 3.5 () Invertible Operators -- 3.5.1 Fredholm Integral Equation -- 3.5.2 Exercises -- 3.5.3 Pointers to Further Study -- 3.6 () Tietze -- 3.6.1 Formulas for an Extension -- 3.6.2 Exercises -- 3.6.3 Pointers to Further Study -- 4 Compact Spaces -- 4.1 Sequentially Compact Spaces -- 4.1.1 Continuous Functions on Sequentially Compact Spaces -- 4.1.2 Bolzano-Weierstrass in Rn -- 4.1.3 Sequentially Compact Sets in Rn -- 4.1.4 Sequentially Compact Sets in Other Spaces -- 4.1.5 The Space C(M) -- 4.1.6 Exercises -- 4.2 The Correct Definition of Compactness -- 4.2.1 Thoughts About the Definition -- 4.2.2 Compact Spaces and Compact Sets -- 4.2.3 Continuous Functions on Compact Spaces -- 4.2.4 Uniform Continuity -- 4.2.5 Exercises -- 4.3 Equivalence of Compactness and Sequential Compactness -- 4.3.1 Relative Compactness -- 4.3.2 Local Compactness -- 4.3.3 Exercises -- 4.4 Finite Dimensional Normed Vector Spaces -- 4.4.1 Exercises -- 4.5 () Ascoli -- 4.5.1 Peano's Existence Theorem.; 4.5.2 Exercises -- 4.5.3 Pointers to Further Study -- 5 Separable Spaces -- 5.1 Dense Subsets of a Metric Space -- 5.1.1 Defining a Vector-Valued Integral -- 5.1.2 Exercises -- 5.2 Separability -- 5.2.1 Second Countability -- 5.2.2 Exercises -- 5.3 () Weierstrass -- 5.3.1 Exercises -- 5.3.2 Pointers to Further Study -- 5.4 () Stone-Weierstrass -- 5.4.1 Exercises -- 5.4.2 Pointers to Further Study -- 6 Properties of Complete Spaces -- 6.1 Cantor's Nested Intersection Theorem -- Notes About Cantor's Theorem -- 6.1.1 Categories -- Thoughts About the Proof -- 6.1.2 Exercises -- 6.2 () Genericity -- 6.2.1 Exercises -- 6.2.2 Pointers to Further Study -- 6.3 () Nowhere Differentiability -- 6.3.1 Exercises -- 6.3.2 Pointers to Further Study -- 6.4 Fixed Points -- 6.4.1 Exercises -- 6.5 () Picard -- 6.5.1 Exercises -- 6.6 () Zeros -- 6.6.1 Exercises -- 6.6.2 Pointers to Further Study -- 6.7 Completion of a Metric Space -- 6.7.1 Other Ways to Complete a Metric Space -- 6.7.2 Exercises -- 7 Connected Spaces -- 7.1 Connectedness -- 7.1.1 Connected Sets -- 7.1.2 Rules for Connected Sets -- 7.1.3 Connected Subsets of R -- 7.1.4 Exercises -- 7.2 Continuous Mappings and Connectedness -- 7.2.1 Continuous Curves -- 7.2.2 Arcwise Connectedness -- 7.2.3 Exiting a Set -- 7.2.4 Exercises -- 7.3 Connected Components -- 7.3.1 Examples of Connected Components -- 7.3.2 Arcwise Connected Components -- 7.3.3 Exercises -- 7.4 () ; Peano -- 7.4.1 Exercises -- 7.4.2 Pointers to Further Study -- Afterword -- Index.