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001445256 001__ 1445256
001445256 003__ OCoLC
001445256 005__ 20230310003820.0
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001445256 019__ $$a1304402670$$a1304463481$$a1304812513
001445256 020__ $$a9783030949464$$q(electronic bk.)
001445256 020__ $$a303094946X$$q(electronic bk.)
001445256 020__ $$z3030949451
001445256 020__ $$z9783030949457
001445256 0247_ $$a10.1007/978-3-030-94946-4$$2doi
001445256 035__ $$aSP(OCoLC)1304358975
001445256 040__ $$aYDX$$beng$$cYDX$$dGW5XE$$dOCLCO$$dEBLCP$$dOCLCF$$dSFB$$dUKAHL$$dOCLCQ
001445256 049__ $$aISEA
001445256 050_4 $$aQA611.28
001445256 08204 $$a514/.325$$223
001445256 1001_ $$aMagnus, Robert,$$eauthor.
001445256 24510 $$aMetric spaces :$$ba companion to analysis /$$cRobert Magnus.
001445256 260__ $$aCham, Switzerland :$$bSpringer,$$c2022.
001445256 300__ $$a1 online resource
001445256 4901_ $$aSpringer undergraduate mathematics series,$$x2197-4144
001445256 500__ $$aIncludes index.
001445256 5050_ $$aIntro -- Preface -- Contents -- Preliminaries on Sets -- Basic Relations -- Basic Operations -- Writing Predicates -- Set-Building Rules -- Relations and Functions -- Cardinals -- Other Notions -- 1 Metric Spaces -- 1.1 Metrics -- 1.1.1 Rationale for Metrics -- 1.1.2 Defining Metric Space -- 1.1.3 Exercises -- 1.2 Examples of Metric Spaces -- 1.2.1 Normed Spaces -- 1.2.2 Subspaces -- 1.2.3 Examples -- Not Subspaces of Normed Spaces -- 1.2.4 Pseudometrics -- 1.2.5 Cauchy-Schwarz, Hölder, Minkowski -- 1.2.6 Exercises -- 1.3 Cantor's Middle Thirds Set -- 1.3.1 Exercises
001445256 5058_ $$a1.4 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
001445256 5058_ $$a2.2.3 Closed Sets -- 2.2.4 Union and Intersection of Closed Sets -- 2.2.5 Characterisation of Open and Closed Sets by Sequences -- 2.2.6 Interior, Closure and Boundary -- 2.2.7 Limit Points of Sets -- 2.2.8 Characterisation of Closure by Limit Points -- 2.2.9 Subspaces -- 2.2.10 Open and Closed Sets in a Subspace -- 2.2.11 Exercises -- 2.3 Continuous Mappings -- 2.3.1 Defining Continuity -- 2.3.2 New Views of Continuity -- 2.3.3 Limits of Functions -- 2.3.4 Characterising Continuity by Sequences -- 2.3.5 Lipschitz Mappings -- 2.3.6 Examples of Continuous Functions -- 2.3.7 Exercises
001445256 5058_ $$a3.2.2 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
001445256 506__ $$aAccess limited to authorized users.
001445256 520__ $$aThis textbook presents the theory of Metric Spaces necessary for studying analysis beyond one real variable. Rich in examples, exercises and motivation, it provides a careful and clear exposition at a pace appropriate to the material. The book covers the main topics of metric space theory that the student of analysis is likely to need. Starting with an overview defining the principal examples of metric spaces in analysis (chapter 1), it turns to the basic theory (chapter 2) covering open and closed sets, convergence, completeness and continuity (including a treatment of continuous linear mappings). There is also a brief dive into general topology, showing how metric spaces fit into a wider theory. The following chapter is devoted to proving the completeness of the classical spaces. The text then embarks on a study of spaces with important special properties. Compact spaces, separable spaces, complete spaces and connected spaces each have a chapter devoted to them. A particular feature of the book is the occasional excursion into analysis. Examples include the Mazurlam theorem, Picard's theorem on existence of solutions to ordinary differential equations, and space filling curves. This text will be useful to all undergraduate students of mathematics, especially those who require metric space concepts for topics such as multivariate analysis, differential equations, complex analysis, functional analysis, and topology. It includes a large number of exercises, varying from routine to challenging. The prerequisites are a first course in real analysis of one real variable, an acquaintance with set theory, and some experience with rigorous proofs.
001445256 588__ $$aOnline resource; title from PDF title page (SpringerLink, viewed March 23, 2022).
001445256 650_0 $$aMetric spaces.
001445256 650_6 $$aEspaces métriques.
001445256 655_7 $$aLlibres electrònics.$$2thub
001445256 655_0 $$aElectronic books.
001445256 77608 $$iPrint version: $$z3030949451$$z9783030949457$$w(OCoLC)1289366162
001445256 830_0 $$aSpringer undergraduate mathematics series,$$x2197-4144
001445256 852__ $$bebk
001445256 85640 $$3Springer Nature$$uhttps://univsouthin.idm.oclc.org/login?url=https://link.springer.com/10.1007/978-3-030-94946-4$$zOnline Access$$91397441.1
001445256 909CO $$ooai:library.usi.edu:1445256$$pGLOBAL_SET
001445256 980__ $$aBIB
001445256 980__ $$aEBOOK
001445256 982__ $$aEbook
001445256 983__ $$aOnline
001445256 994__ $$a92$$bISE