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**Author(s):** Thomas Mack

**Publisher:** The MIT Press

**Year:** 2023

**Language:** English

Contents

Introduction

1. Group Theory

1.1 Definitions and Examples

1.2 Subgroups and Group Homomorphisms

1.3 Group Constructions

1.4 The Isomorphism Theorems

1.5 Group Actions

1.6 Cyclic Groups

1.7 Permutation Groups

1.8 p-Groups and the Sylow Theorems

1.9 Solvable and Nilpotent Groups

1.10 Free Groups and Presentations

1.11 Further Topics

1.12 Further Reading

Exercises

2. Commutative Algebra

2.1 Rings

2.2 Ideals

2.3 Polynomials

2.4 Modules

2.5 Module Constructions

2.6 Noetherian Modules

2.7 Prime and Maximal Ideals

2.8 Localization

2.9 Gauss’s Lemma

2.10 Principal Ideal Domains

2.11 Field Extensions

2.12 Finite Fields

2.13 Further Topics

2.14 Further Reading

Exercises

3. Linear Algebra

3.1 Vector Spaces

3.2 Dimension

3.3 Vector Space Constructions

3.4 Eigenvalues and Eigenvectors

3.5 The Determinant

3.6 Matrices

3.7 Matrix Operations

3.8 Inner Product Spaces

3.9 Matrix Decompositions

3.10 Further Topics

3.11 Further Reading

Exercises

4. Topology

4.1 Definitions and Examples

4.2 Continuity

4.3 Topological Space Constructions

4.4 Separation Axioms

4.5 Connectedness

4.6 Compactness

4.7 Tychonoff’s Theorem

4.8 Metric Spaces

4.9 Completeness

4.10 Homotopy

4.11 Further Topics

4.12 Further Reading

Exercises

5. Real Analysis

5.1 Limits

5.2 Infinite Series

5.3 Uniform Convergence

5.4 Differentiation on R

5.5 Taylor’s Theorem

5.6 Measurable Spaces

5.7 Measurable Functions

5.8 Integration

5.9 Measure Extensions

5.10 Borel Measure

5.11 The Fundamental Theorem of Calculus

5.12 Further Topics

5.13 Further Reading

Exercises

6. Multivariable Analysis

6.1 Multivariable Differentiation

6.2 Multivariable Integration

6.3 The Change of Variables Formula

6.4 Differential Equations

6.5 Common Derivatives and Integrals

6.6 The Gaussian Integral

6.7 The Weierstrass Approximation Theorem

6.8 The Constant Rank Theorem

6.9 Further Topics

6.10 Further Reading

Exercises

7. Complex Analysis

7.1 Contour Integrals

7.2 The Jordan Curve Theorem

7.3 The Topology of Contours

7.4 Green’s Theorem

7.5 The Cauchy–Riemann Equations

7.6 Cauchy’s Integral Formula

7.7 Consequences of Cauchy’s Integral Formula

7.8 Meromorphic Functions

7.9 Residues

7.10 The Open Mapping Theorem

7.11 Tauberian Theorems

7.12 Further Topics

7.13 Further Reading

Exercises

8. Number Theory

8.1 Galois Theory

8.2 Algebraic Integers

8.3 Prime Factorization in Ok

8.4 Quadratic Fields

8.5 Cyclotomic Extensions

8.6 Diophantine Equations

8.7 Quadratic Reciprocity

8.8 Solvability by Radicals

8.9 The Riemann ζ-Function

8.10 The Prime Number Theorem

8.11 Further Topics

8.12 Further Reading

Exercises

9. Probability

9.1 Definitions and Constructions

9.2 Densities

9.3 Lp spaces

9.4 The Radon–Nikodym Theorem

9.5 Mean and Variance

9.6 Joint Density Functions

9.7 Common Probability Distributions

9.8 Convergence of Distributions

9.9 Higher Moments and Characteristic Functions

9.10 The Central Limit Theorem

9.11 Further Topics

9.12 Further Reading

Exercises

Appendix

A.1 Set Theory

A.2 The Axiom of Choice

A.3 Cardinality

A.4 Real and Complex Numbers

References

List of Symbols

Index

Introduction

1. Group Theory

1.1 Definitions and Examples

1.2 Subgroups and Group Homomorphisms

1.3 Group Constructions

1.4 The Isomorphism Theorems

1.5 Group Actions

1.6 Cyclic Groups

1.7 Permutation Groups

1.8 p-Groups and the Sylow Theorems

1.9 Solvable and Nilpotent Groups

1.10 Free Groups and Presentations

1.11 Further Topics

1.12 Further Reading

Exercises

2. Commutative Algebra

2.1 Rings

2.2 Ideals

2.3 Polynomials

2.4 Modules

2.5 Module Constructions

2.6 Noetherian Modules

2.7 Prime and Maximal Ideals

2.8 Localization

2.9 Gauss’s Lemma

2.10 Principal Ideal Domains

2.11 Field Extensions

2.12 Finite Fields

2.13 Further Topics

2.14 Further Reading

Exercises

3. Linear Algebra

3.1 Vector Spaces

3.2 Dimension

3.3 Vector Space Constructions

3.4 Eigenvalues and Eigenvectors

3.5 The Determinant

3.6 Matrices

3.7 Matrix Operations

3.8 Inner Product Spaces

3.9 Matrix Decompositions

3.10 Further Topics

3.11 Further Reading

Exercises

4. Topology

4.1 Definitions and Examples

4.2 Continuity

4.3 Topological Space Constructions

4.4 Separation Axioms

4.5 Connectedness

4.6 Compactness

4.7 Tychonoff’s Theorem

4.8 Metric Spaces

4.9 Completeness

4.10 Homotopy

4.11 Further Topics

4.12 Further Reading

Exercises

5. Real Analysis

5.1 Limits

5.2 Infinite Series

5.3 Uniform Convergence

5.4 Differentiation on R

5.5 Taylor’s Theorem

5.6 Measurable Spaces

5.7 Measurable Functions

5.8 Integration

5.9 Measure Extensions

5.10 Borel Measure

5.11 The Fundamental Theorem of Calculus

5.12 Further Topics

5.13 Further Reading

Exercises

6. Multivariable Analysis

6.1 Multivariable Differentiation

6.2 Multivariable Integration

6.3 The Change of Variables Formula

6.4 Differential Equations

6.5 Common Derivatives and Integrals

6.6 The Gaussian Integral

6.7 The Weierstrass Approximation Theorem

6.8 The Constant Rank Theorem

6.9 Further Topics

6.10 Further Reading

Exercises

7. Complex Analysis

7.1 Contour Integrals

7.2 The Jordan Curve Theorem

7.3 The Topology of Contours

7.4 Green’s Theorem

7.5 The Cauchy–Riemann Equations

7.6 Cauchy’s Integral Formula

7.7 Consequences of Cauchy’s Integral Formula

7.8 Meromorphic Functions

7.9 Residues

7.10 The Open Mapping Theorem

7.11 Tauberian Theorems

7.12 Further Topics

7.13 Further Reading

Exercises

8. Number Theory

8.1 Galois Theory

8.2 Algebraic Integers

8.3 Prime Factorization in Ok

8.4 Quadratic Fields

8.5 Cyclotomic Extensions

8.6 Diophantine Equations

8.7 Quadratic Reciprocity

8.8 Solvability by Radicals

8.9 The Riemann ζ-Function

8.10 The Prime Number Theorem

8.11 Further Topics

8.12 Further Reading

Exercises

9. Probability

9.1 Definitions and Constructions

9.2 Densities

9.3 Lp spaces

9.4 The Radon–Nikodym Theorem

9.5 Mean and Variance

9.6 Joint Density Functions

9.7 Common Probability Distributions

9.8 Convergence of Distributions

9.9 Higher Moments and Characteristic Functions

9.10 The Central Limit Theorem

9.11 Further Topics

9.12 Further Reading

Exercises

Appendix

A.1 Set Theory

A.2 The Axiom of Choice

A.3 Cardinality

A.4 Real and Complex Numbers

References

List of Symbols

Index