Preface
For students: How to use this book
Chapter 1. Intervals
1.1. Distance and neighborhoods
1.2. Interior of a set
Chapter 2. Topology of the real line
2.1. Open subsets of \R
2.2. Closed subsets of \R
Chapter 3. Continuous functions from \R to \R
3.1. Continuity—as a local property
3.2. Continuity—as a global property
3.3. Functions defined on subsets of \R
Chapter 4. Sequences of real numbers
4.1. Convergence of sequences
4.2. Algebraic combinations of sequences
4.3. Sufficient condition for convergence
4.4. Subsequences
Chapter 5. Connectedness and the intermediate value theorem
5.1. Connected subsets of \R
5.2. Continuous images of connected sets
5.3. Homeomorphisms
Chapter 6. Compactness and the extreme value theorem
6.1. Compactness
6.2. Examples of compact subsets of \R
6.3. The extreme value theorem
Chapter 7. Limits of real valued functions
7.1. Definition
7.2. Continuity and limits
Chapter 8. Differentiation of real valued functions
8.1. The families \lobo𝑂 and \lobo𝑜
8.2. Tangency
8.3. Linear approximation
8.4. Differentiability
Chapter 9. Metric spaces
9.1. Definitions
9.2. Examples
9.3. Equivalent metrics
Chapter 10. Interiors, closures, and boundaries
10.1. Definitions and examples
10.2. Interior points
10.3. Accumulation points and closures
Chapter 11. The topology of metric spaces
11.1. Open and closed sets
11.2. The relative topology
Chapter 12. Sequences in metric spaces
12.1. Convergence of sequences
12.2. Sequential characterizations of topological properties
12.3. Products of metric spaces
Chapter 13. Uniform convergence
13.1. The uniform metric on the space of bounded functions
13.2. Pointwise convergence
Chapter 14. More on continuity and limits
14.1. Continuous functions
14.2. Maps into and from products
14.3. Limits
Chapter 15. Compact metric spaces
15.1. Definition and elementary properties
15.2. The extreme value theorem
15.3. Dini’s theorem
Chapter 16. Sequential characterization of compactness
16.1. Sequential compactness
16.2. Conditions equivalent to compactness
16.3. Products of compact spaces
16.4. The Heine–Borel theorem
Chapter 17. Connectedness
17.1. Connected spaces
17.2. Arcwise connected spaces
Chapter 18. Complete spaces
18.1. Cauchy sequences
18.2. Completeness
18.3. Completeness vs. compactness
Chapter 19. A fixed point theorem
19.1. The contractive mapping theorem
19.2. Application to integral equations
Chapter 20. Vector spaces
20.1. Definitions and examples
20.2. Linear combinations
20.3. Convex combinations
Chapter 21. Linearity
21.1. Linear transformations
21.2. The algebra of linear transformations
21.3. Matrices
21.4. Determinants
21.5. Matrix representations of linear transformations
Chapter 22. Norms
22.1. Norms on linear spaces
22.2. Norms induce metrics
22.3. Products
22.4. The space \fml𝐵(𝑆,𝑉)
Chapter 23. Continuity and linearity
23.1. Bounded linear transformations
23.2. The Stone–Weierstrass theorem
23.3. Banach spaces
23.4. Dual spaces and adjoints
Chapter 24. The Cauchy integral
24.1. Uniform continuity
24.2. The integral of step functions
24.3. The Cauchy integral
Chapter 25. Differential calculus
25.1. \lobo𝑂 and \lobo𝑜 functions
25.2. Tangency
25.3. Differentiation
25.4. Differentiation of curves
25.5. Directional derivatives
25.6. Functions mapping into product spaces
Chapter 26. Partial derivatives and iterated integrals
26.1. The mean value theorem(s)
26.2. Partial derivatives
26.3. Iterated integrals
Chapter 27. Computations in \Rⁿ
27.1. Inner products
27.2. The gradient
27.3. The Jacobian matrix
27.4. The chain rule
Chapter 28. Infinite series
28.1. Convergence of series
28.2. Series of positive scalars
28.3. Absolute convergence
28.4. Power series
Chapter 29. The implicit function theorem
29.1. The inverse function theorem
29.2. The implicit function theorem
Chapter 30. Higher order derivatives
30.1. Multilinear functions
30.2. Second order differentials
30.3. Higher order differentials
Appendix A. Quantifiers
Appendix B. Sets
Appendix C. Special subsets of \R
Appendix D. Logical connectives
D.1. Disjunction and conjunction
D.2. Implication
D.3. Restricted quantifiers
D.4. Negation
Appendix E. Writing mathematics
E.1. Proving theorems
E.2. Checklist for writing mathematics
E.3. Fraktur and Greek alphabets
Appendix F. Set operations
F.1. Unions
F.2. Intersections
F.3. Complements
Appendix G. Arithmetic
G.1. The field axioms
G.2. Uniqueness of identities
G.3. Uniqueness of inverses
G.4. Another consequence of uniqueness
Appendix H. Order properties of \R
Appendix I. Natural numbers and mathematical induction
Appendix J. Least upper bounds and greatest lower bounds
J.1. Upper and lower bounds
J.2. Least upper and greatest lower bounds
J.3. The least upper bound axiom for \R
J.4. The Archimedean property
Appendix K. Products, relations, and functions
K.1. Cartesian products
K.2. Relations
K.3. Functions
Appendix L. Properties of functions
L.1. Images and inverse images
L.2. Composition of functions
L.3. The identity function
L.4. Diagrams
L.5. Restrictions and extensions
Appendix M. Functions that have inverses
M.1. Injections, surjections, and bijections
M.2. Inverse functions
Appendix N. Products
Appendix O. Finite and infinite sets
Appendix P. Countable and uncountable sets
Bibliography
Index