Analysis I

Preface
Contents
Chapter I Foundations
1 Fundamentals of Logic
2 Sets
Elementary Facts
The Power Set
Complement, Intersection and Union
Products
Families of Sets
3 Functions
Simple Examples
Composition of Functions
Commutative Diagrams
Injections, Surjections and Bijections
Inverse Functions
Set Valued Functions
4 Relations and Operations
Equivalence Relations
Order Relations
Operations
5 The Natural Numbers
The Peano Axioms
The Arithmetic of Natural Numbers
The Division Algorithm
The Induction Principle
Recursive Definitions
6 Countability
Permutations
Equinumerous Sets
Countable Sets
Infinite Products
7 Groups and Homomorphisms
Groups
Subgroups
Cosets
Homomorphisms
Isomorphisms
8 Rings, Fields and Polynomials
Rings
The Binomial Theorem
The Multinomial Theorem
Fields
Ordered Fields
Formal Power Series
Polynomials
Polynomial Functions
Division of Polynomials
Linear Factors
Polynomials in Several Indeterminates
9 The Rational Numbers
The Integers
The Rational Numbers
Rational Zeros of Polynomials
Square Roots
10 The Real Numbers
Order Completeness
Dedekind’s Construction of the Real Numbers
The Natural Order on R
The Extended Number Line
A Characterization of Supremum and Infimum
The Archimedean Property
The Density of the Rational Numbers in R
nth Roots
The Density of the Irrational Numbers in R
Intervals
11 The Complex Numbers
Constructing the Complex Numbers
Elementary Properties
Computation with Complex Numbers
Balls in K
12 Vector Spaces, Affine Spaces and Algebras
Vector Spaces
Linear Functions
Vector Space Bases
Affine Spaces
Affine Functions
Polynomial Interpolation
Algebras
Difference Operators and Summation Formulas
Newton Interpolation Polynomials
Chapter II Convergence
1 Convergence of Sequences
Sequences
Metric Spaces
Cluster Points
Convergence
Bounded Sets
Uniqueness of the Limit
Subsequences
2 Real and Complex Sequences
Null Sequences
Elementary Rules
The Comparison Test
Complex Sequences
3 Normed Vector Spaces
Norms
Balls
Bounded Sets
Examples
The Space of Bounded Functions
Inner Product Spaces
The Cauchy-Schwarz Inequality
Euclidean Spaces
Equivalent Norms
Convergence in Product Spaces
4 Monotone Sequences
Bounded Monotone Sequences
Some Important Limits
5 Infinite Limits
Convergence to ±∞
The Limit Superior and Limit Inferior
The Bolzano-Weierstrass Theorem
6 Completeness
Cauchy Sequences
Banach Spaces
Cantor’s Construction of the Real Numbers
7 Series
Convergence of Series
Harmonic and Geometric Series
Calculating with Series
Convergence Tests
Alternating Series
Decimal, Binary and Other Representations of Real Numbers
The Uncountability of R
8 Absolute Convergence
Majorant, Root and Ratio Tests
The Exponential Function
Rearrangements of Series
Double Series
Cauchy Products
9 Power Series
The Radius of Convergence
Addition and Multiplication of Power Series
The Uniqueness of Power Series Representations
Chapter III Continuous Functions
1 Continuity
Elementary Properties and Examples
Sequential Continuity
Addition and Multiplication of Continuous Functions
One-Sided Continuity
2 The Fundamentals of Topology
Open Sets
Closed Sets
The Closure of a Set
The Interior of a Set
The Boundary of a Set
The Hausdorff Condition
Examples
A Characterization of Continuous Functions
Continuous Extensions
Relative Topology
General Topological Spaces
3 Compactness
Covers
A Characterization of Compact Sets
Sequential Compactness
Continuous Functions on Compact Spaces
The Extreme Value Theorem
Total Boundedness
Uniform Continuity
Compactness in General Topological Spaces
4 Connectivity
Definition and Basic Properties
Connectivity in R
The Generalized Intermediate Value Theorem
Path Connectivity
Connectivity in General Topological Spaces
5 Functions on R
Bolzano’s Intermediate Value Theorem
Monotone Functions
Continuous Monotone Functions
6 The Exponential and Related Functions
Euler’s Formula
The Real Exponential Function
The Logarithm and Power Functions
The Exponential Function on iR
The Definition of π and its Consequences
The Tangent and Cotangent Functions
The Complex Exponential Function
Polar Coordinates
Complex Logarithms
Complex Powers
A Further Representation of the Exponential Function
Chapter IV Differentiation in One Variable
1 Differentiability
The Derivative
Linear Approximation
Rules for Differentiation
The Chain Rule
Inverse Functions
Differentiable Functions
Higher Derivatives
One-Sided Differentiability
2 The Mean Value Theorem and its Applications
Extrema
The Mean Value Theorem
Monotonicity and Differentiability
Convexity and Differentiability
The Inequalities of Young, Hölder and Minkowski
The Mean Value Theorem for Vector Valued Functions
The Second Mean Value Theorem
L’Hospital’s Rule
3 Taylor’s Theorem
The Landau Symbol
Taylor’s Formula
Taylor Polynomials and Taylor Series
The Remainder Function in the Real Case
Polynomial Interpolation
Higher Order Difference Quotients
4 Iterative Procedures
Fixed Points and Contractions
The Banach Fixed Point Theorem
Newton’s Method
Chapter V Sequences of Functions
1 Uniform Convergence
Pointwise Convergence
Uniform Convergence
Series of Functions
The Weierstrass Majorant Criterion
2 Continuity and Differentiability for Sequences of Functions
Continuity
Locally Uniform Convergence
The Banach Space of Bounded Continuous Functions
Differentiability
3 Analytic Functions
Differentiability of Power Series
Analyticity
Antiderivatives of Analytic Functions
The Power Series Expansion of the Logarithm
The Binomial Series
The Identity Theorem for Analytic Functions
4 Polynomial Approximation
Banach Algebras
Density and Separability
The Stone-Weierstrass Theorem
Trigonometric Polynomials
Periodic Functions
The Trigonometric Approximation Theorem
Appendix Introduction to Mathematical Logic
Bibliography
Index