文档介绍：Companion to Real Analysis
John M. Erdman
Portland State University
Version September 27, 2009
c 2007 John M. Erdman
E-mail address: ******@
Contents
PREFACE vii
Chapter 1. SETS 1
. Set Notation 1
. Families of Sets 1
. Subsets 2
. Unions and Intersections 2
. Complements 3
. Symmetric Difference 4
. Notation for Sets of Numbers 5
Chapter 2. FUNCTIONS 7
. Cartesian Products 7
. Relations 7
. Functions 8
. Images and Inverse Images 9
. Composition of Functions 9
. The Identity Function 9
. Diagrams 10
. Some Special Functions 10
. Injections, Surjections, and Bijections 11
. Inverse Functions 13
. Equivalence Relations and Quotients 14
Chapter 3. CARDINALITY 17
. Finite and Infinite Sets 17
. Countable and Uncountable Sets 18
Chapter 4. GROUPS, VECTOR SPACES, AND ALGEBRAS 21
. Operations 21
. Groups 22
. Homomorphisms of Semigroups and Groups 24
. Vector Spaces 25
. Linear Transformations 26
. Rings and Algebras 28
. Ring and Algebra Homomorphisms 30
Chapter 5. PARTIALLY ORDERED SETS 33
. Partial and Linear Orderings 33
. Infima and Suprema 34
. Zorn’s Lemma 35
. Lattices 37
. Lattice Homomorphisms 39
. Boolean Algebras 39
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Chapter 6. THE REAL NUMBERS 43
. Axioms Defining the Real Numbers 43
. Construction of the Real Numbers 44
. Elementary Functions 47
. Absolute Value 47
. Some Useful Inequalities 48
. Complex Numbers 49
Chapter 7. SEQUENCES AND INDEXED FAMILIES 51
. Sequences 51
. Indexed Families of Sets 52
. Limit Inferior and Limit Superior (for Sets) 53
. Limit Inferior and Limit Superior (for Real Numbers) 54
. Subsequences and Cluster Points 58
Chapter 8. CATEGORIES 59
. Objects and Morphisms 59
. Quotients 61
. Products 63
. Coproducts 65
Chapter 9. ORDERED VECTOR SPACES 67
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