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Mathematical Analysis: A Concise Introduction

Bernd S. W. Schröder
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This is the Greatest Hits version of mathematical analysis. It covers all the high points of real and abstract analysis, but doesn't go into depth on anything. The treatment is rigorous and error-free. Roughly half the book deals with real analysis theorems, and half with abstract analysis and function spaces (including multivariable calculus). There is also a short section on physical applications, dealing mostly with differential equations.

The book skimps on worked examples. Most of the illustrative examples are in the exercises, often with hints. Many traditional analysis topics appear only in the exercises, for example, Stirling's formula for n!, Riemann-Stieltjes integrals, Lagrange multipliers, and the Stone-Weierstrass Theorem,

It may seem peculiar to subtitle a 562-page book "concise", but that is an accurate summary of the approach. The book manages to cover an enormous amount of material in those pages.

As I was reading I was continually puzzling over where the book would fit in the curriculum. The author used the draft in a two-quarter course at his university, but it must have been a struggle to get through all the material in that time. In some ways the book looks like a text for an introduction to proofs course, but it also includes a number of topics that would normally be in graduate courses, and everything in between. Is this "The Only Analysis Book You'll Ever Need"? Probably not — there's no complex analysis, for example, and Fourier analysis is only touched on briefly.

My big gripe with this book is that it is uninspiring. It never shows you why people get excited about analysis. The history of real analysis is largely a history of pathological cases and counterexamples, and how analysts prevailed over these difficulties to create a beautiful theory. This book shows you the theory but not the beauty.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.



1. The Real Numbers.

1.1 Field Axioms.

1.2 Order Axioms.

1.3 Lowest Upper and Greatest Lower Bounds.

1.4 Natural Numbers, Integers and Rational Numbers.

1.5 Recursion, Induction, Summations and Products.

2. Sequences of Real Numbers.

2.1 Limits.

2.2 Limit Laws.

2.3 Cauchy Sequences.

2.4 Bounded Sequences.

2.5 Infinite Limits.

3. Continuous Functions.

3.1 Limits of Functions.

3.2 Limit Laws.

3.3 One-Sided Limits and Infinite Limits.

3.4 Continuity.

3.5 Properties of Continuous Functions.

3.6 Limits at Infinity.

4. Differentiable Functions.

4.1 Differentiability.

4.2 Differentiation Rules.

4.3 Rolle's Theorem and the Mean Value Theorem.

5. The Riemann Integral I.

5.1 Riemann Sums and the Integral.

5.2 Uniform Continuity and Integrability of Continuous Functions.

5.3 The Fundamental Theorem of Calculus.

5.4 The Darboux Integral.

6. Series of Real Numbers I.

6.1 Series as a Vehicle to Define Infinite Sums.

6.2 Absolute Convergence and Unconditional Convergence.

7. Some Set Theory.

7.1 The Algebra of Sets.

7.2 Countable Sets.

7.3 Uncountable Sets.

8. The Riemann Integral II.

8.1 Outer Lebesgue Measure.

8.2 Lebesgue's Criterion for Riemann Integrability.

8.3 More Integral Theorems.

8.4 Improper Riemann Integrals.

9. The Lebesgue Integral.

9.1 Lebesgue Measurable Sets.

9.2 Lebesgue Measurable Functions.

9.3 Lebesgue Integration.

9.4 Lebesgue Integrals vs. Riemann Integrals.

10. Series of Real Numbers II.

10.1 Limits Superior and Inferior.

10.2 The Root Test and the Ratio Test.

10.3 Power Series.

11. Sequences of Functions.

11.1 Notions of Convergence.

11.2 Uniform Convergence.

12. Transcendental Functions.

12.1 The Exponential Function.

12.2 Sine and Cosine.

12.3 L?H?opital's Rule.

13. Numerical Methods 203.

13.1 Approximation with Taylor Polynomials.

13.2 Newton's Method.

13.3 Numerical Integration.


14. Integration on Measure Spaces.

14.1 Measure Spaces.

14.2 Outer Measures.

14.3 Measurable Functions.

14.4 Integration of Measurable Functions.

14.5 Monotone and Dominated Convergence.

14.6 Convergence in Mean, in Measure and Almost Everywhere.

14.7 Product _-Algebras.

14.8 Product Measures and Fubini's Theorem.

15. The Abstract Venues for Analysis.

15.1 Abstraction I: Vector Spaces.

15.2 Representation of Elements: Bases and Dimension.

15.3 Identification of Spaces: Isomorphism.

15.4 Abstraction II: Inner Product Spaces.

15.5 Nicer Representations: Orthonormal Sets.

15.6 Abstraction III: Normed Spaces.

15.7 Abstraction IV: Metric Spaces.

15.8 L p Spaces.

15.9 Another Number Field: Complex Numbers.

16. The Topology of Metric Spaces.

16.1 Convergence of Sequences.

16.2 Completeness.

16.3 Continuous Functions.

16.4 Open and Closed Sets.

16.5 Compactness.

16.6 The Normed Topology of Rd.

16.7 Dense Subspaces.

16.8 Connectedness.

16.9 Locally Compact Spaces.

17. Differentiation in Normed Spaces.

17.1 Continuous Linear Functions.

17.2 Matrix Representation of Linear Functions.

17.3 Differentiability.

17.4 The Mean Value Theorem.

17.5 How Partial Derivatives Fit In.

17.6 Multilinear Functions (Tensors) .

17.7 Higher Derivatives.

17.8 The Implicit Function Theorem.

18. Measure, Topology and Differentiation.

18.1 Lebesgue Measurable Sets in Rd.

18.2 C1 and Approximation of Integrable Functions.

18.3 Tensor Algebra and Determinants.

18.4 Multidimensional Substitution.

19. Manifolds and Integral Theorems.

19.1 Manifolds.

19.2 Tangent Spaces and Differentiable Functions.

19.3 Differential Forms, Integrals over The Unit Cube.

19.4 k-Forms and Integrals over k-Chains.

19.5 Integration on Manifolds.

19.6 Stokes' Theorem.

20. Hilbert Spaces.

20.1 Orthonormal Bases.

20.2 Fourier Series.

20.3 The Riesz Representation Theorem.


21. Physics Background.

21.1 Harmonic Oscillation.

21.2 Heat and Diffusion.

21.3 Separation of Variables, Fourier Series and Ordinary Differential Equations.

21.4 Maxwell's Equations.

21.5 The Navier Stokes Equation for the Conservation of Mass.

22. Ordinary Differential Equations.

22.1 Banach Space Valued Differential Equations.

22.2 An Existence and Uniqueness Theorem.

22.3 Linear Differential Equations.

23. The Finite Element Method.

23.1 Ritz-Galerkin Approximation.

23.2 Weakly Differentiable Functions.

23.3 Sobolev Spaces.

23.4 Elliptic Differential Operators.

23.5 Finite Elements.

Conclusion and Outlook.


A. Logic.

A.1 Statements.

A.2 Negations.

B. Set Theory.

B.1 The Zermelo-Fraenkel Axioms.

B.2 Relations and Functions.

C. Natural Numbers, Integers and Rational Numbers.

C.1 The Natural Numbers.

C.2 The Integers.

C.3 The Rational Numbers.