Elements of abstract analysis抽象分析原理

內容概要

“an interesting mix of algebra，topology and functional analysis， with chapter titles like Algebraic Structure，Linear Structure， Geometric Structure and Topological Structure。”　　“Michael OSearcoids book Element of Abstract Analysis is a book that deals with the fundamental principles of mathematics in a broad framework of set-theoretic approach。The book is written with a deep insight。The contents are conceptually useful to students。The book is a nice readable treat for every student of mathematics and is highly recommended for one semester course at M.Sc。 level?！?　　“This book covers the elements of abstract analysis，treated mostly within the context of topological spaces。This is a carefully written book，starting from set-theoretic axioms，and containing about 180 problems and exercises with hints and solutions。It is an ideal source of information for independent student study。There is a comprehensive index?！?　　“The book presents some concepts of analysis and the mathematical structures which enfold them。the reviewer recommends the book to everyone interested in analysis，especially those scholars interested in studying more advanced areas like Functional Analysis，or Operator Theory?！?　　“The book is written specifically for final-year undergraduate students who should already be familiar with most of the mathematical structures discussed。It reviews the concepts at a slightly greater level of abstraction and enables students to understand their place within the broad framework of set-based mathematics。This book is a rigorous，self-contained introduction to functional analysis that will also serve as a text on abstract mathematics?！?　　“This is a book about a few elementary concepts of analysis and the mathematical structures which enfold them。The book is written specifically for final-year and undergraduate students who should already be familiar with most of the mathematical structures discussed  and with many of the principal analytical concepts 。It reviews the concepts at a slightly greater level of abstraction and enables students to understand their place within the broad framework of set-based mathematics?！?　　While there are many books on functional analysis，Elements of Abstract Analysis takes a very different approach。Unlike other books， it provides a comprehensive overview of the elementary concepts of analysis while preparing students to cross the threshold of functional analysis?！　he book is written specifically for final-year undergraduate students who should already be familiar with most of the mathematical structures discussed for example，rings，linear spaces，and metric spaces - and with many of the principal analytical concepts - convergence，connectedness，continuity，compactness and completeness。 It reviews the concepts at a slightly greater level of abstraction and enables students to understand their place within the broad framework of set-based mathematics?！　arefully crafted，clearly written and precise，and with numerous exercises and examples，Elements of Abstract Analysis is a rigorous， self-contained introduction to functional analysis that will also serve as a text on abstract mathematics。

書籍目錄

Let US from point to pomt this story knowTo make the even truth in pleasure flowAll’s Well that Ends WellList of Symbols1.Sets  1.1  Set Theory　1.2  Relations and Functions  1.3  Ordered Sets  1.4　Ordinals  1.5　The Axiom of Choice2.Counting  2.1　Counting Numbers  2.2　Cardinality  2.3　Enumeration  2.4  Cardinality of Unions and Products3.Algebraic Structure　3.1　Elementary Algebraic Structures　3.2　Vector Spaces　3.3　Algebras　3.4　Preservation of Algebraic StructureAnalytic Structure　4.1　Ordered Algebraic Structure　4.2　Number Systems　4.3  Real and Complex Functions　4.4　Inequalities5.Linear Structure  5.1  Linear Spaces and Algebras.  5.2  Linear Shapes  5.3  Linear Functionals6.Geometric Structure  6.1  Semimetrics and Metrics  6.2  Seminorms and Norms  6.3  Sesquilinear Forms and Inner Products7.Topological Structure  7.1  Topologies  7.2  NeighbOurh00ds  7.3  Cardinality and Topology  7.4  Separation8.Continuity and Openness  8.1  Preservation of Topological Structure  8.2  Topologies Defined by Functions  8.3  Derived Topological Spaces  8.4  Topologies on Linear Spaces9. C0nnectedness  9.1  Connected Spaces  9.2  Pathwise Connectedness10.Convergence  10.1  Filters  10.2  Limits11.Compactness.  11.1  Compact Topological Spaces  11.2  Compact Hausdorff Spaces  11.3  Local Compactness12.Completeness  12.1  Complete Metric Spaces  12.2  Banach Spaces  12.3  Hilbert Spaces  12.4  Banach AlgebrasSolutionsBibliographyIndex

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