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Analysis

Real Analysis

Proofs books

These aren’t actually analysis books, but in the math curriculum that’s often where they get used, since intro to real analysis is often the place where math students start to write (and really have to understand) proofs.

• Velleman, How to Prove It (2e, 1e)

  A series of blog posts discussing solutions to Velleman’s exercises: http://technotes-himanshu.blogspot.co.uk/search/label/htpi

• Solow, How to Read and Do Proofs: An Introduction to Mathematical Thought Processes (6e, 5e, 4e)

Introductory analysis

These books ease you into real analysis. They are designed to address a difficulty in the evolving US math curriculum, namely that because rigorous calculus textbooks (like Spivak and Apostol, or those books titled “Advanced Calculus”) are no longer commonly used in lower-level courses, students arrive in real analysis courses without knowing how to read and write proofs or understanding the theoretical foundation of calculus. These books help you with understanding how to do proofs, and they feed you the information in a way that’s easier to digest. They don’t cover all the material you need, and thus they aren’t sufficient on their own to master the subject, but they can make the other books (which I call “standard” below) more manageable.

All of these books stop short of Lebesgue integration, so they can’t be considered complete introductions to real analysis.

• Alcock, How to Think About Analysis (1e)

  This book begins with a discussion of how to read a math book. This might seem like a trivial thing to discuss, but it is a real problem for many people because math books are presented in a way that requires a lot of engagement on the part of the reader. Many students are not prepared when they encounter a book that is full of proofs and not much else. Alcock covers topics like how to relate theorems to definitions, and even how to study. The remainder of the book introduces major topics in analysis: sequences, series, continuity, differentiability, integrability, the real numbers. This book is written in such an elementary manner that it has a bit of a “for dummies” feel to it (though it’s never condescening or childish), but if you can swallow your pride, there’s a lot of good information here.

• Schramm, Introduction to Real Analysis (Dover)

  Schramm begins with a section on how to write proofs. What’s really notable about the rest of the book is that he discusses how a number of major theorems about the real numbers are essentially equivalent, and he shows you how to establish this equivalence. He wraps up by constructing the real number system via Dedekind cuts, which is a sensible transition to a book like Rudin that begins with Dedekind cuts. Every section is followed by a fair number of problems: they aren’t particularly difficult for the most part, but they do a good job of seeing that you understand the material.

• Ross, Elementary Analysis: The Theory of Calculus (2e, 1e)

  Ross eases you into real analysis by putting extra emphasis on the fundamental topics. The book uses a decent number of illustrations when discussing basic topology.

• Abbot, Understanding Analysis (1e paperback)

  Abbot’s book is more a traditional real analysis book that the previous ones in this section. It doesn’t include chapters teaching you how to do mathemetics, it just gives good, clear explanations of the material.

• Rosenlicht. Introduction to Analysis (1e)

  A classic book (1968), well-regarded for its clarity.

Standard undergraduate analysis

These books present the standard undergraduate course in real analysis. They include the Lebesgue integral, but stop short of measure theory.

• Rudin, Principles of Mathematical Analysis (3e intl, 3e intl PIE)

  AKA “Baby Rudin”. The definitive book on real analysis. For some, it’s the yardstick by which all other real analysis textbooks are measured, and found wanting. (For others, it’s a work of elitist arrogance designed to make you feel inferior to the author.) Rudin expects you to keep up with him, and makes no apologies if you don’t. The book is carefully constructed, though, so that at each step you have the definitions and theorems needed to proceed. This approach requires a lot of work from you as reader, but it does pay off. His exercises are demanding, but they force you to engage the material on another level and really understand it.

  If you don’t have experience reading serious math books, one of the “hand-holding” books mentioned above is strongly suggested. Those authors no doubt had this book in mind as one of the texts that they intended to prepare you for.

  Most students (especially those not taking a university course) will use some other textbook along with Rudin. Pretty much any of the other books mentioned here makes a suitable counterpoint to Rudin’s style and fills in many of the details that he glosses over.

  In struggling with Rudin, I found various supplements and partial solutions manuals online. Of these, most were not very helpful; often, I had actually worked out the same proof that the they showed, but I didn’t understand how it served to prove the premise and the solution came with no explanation. (That’s still some help; though: if I have some reassurance that I’ve seen the statement that’s necessary for the proof, then I know I’m on the right track.)

  The one resource that I did find most helpful is George Bergman’s “Supplements”, which has commentary on Rudin’s text as well as the exercises [https://math.berkeley.edu/~gbergman/ug.hndts/m104_Rudin_exs.pdf]. Bergman also wrote up his own errata listing things he had asked Rudin to change: [https://math.berkeley.edu/~gbergman/ug.hndts/m104_Rudin_notes.pdf].

  UW-Madison hosts a solution manual by Roger Cooke, [http://minds.wisconsin.edu/handle/1793/67009]; this wasn’t always thorough enough, but it was of some help.

• Apostol, Mathematical Analysis (2e)

  Another classic text (1974), with more extensive explanations than Rudin.

  This page has some solutions for exercises: http://www.csie.ntu.edu.tw/~b89089/solution.html

• Pugh, Real Mathematical Analysis (2e, 1e)

  Pugh’s book is probably the main competitor to Baby Rudin among more recent books. Notable for its clear explanations and its many exercises. It contains many illustrations, but I find most of them to be poorly chosen, as if the author recognizes that illustrations are good to have but doesn’t grasp how to use them to illustrate concepts. This isn’t much of a con, just puzzling.

• Stromberg, An Introduction to Classical Real Analysis (AMS 2015 reissue, Wadsworth 1981 original)

  Well-liked for its topic selection and exercises. Exercise difficulty scales more smoothly than Rudin’s. The AMS edition is very nice, and probably almost as cheap new as a used copy of the Wadsworth will be.

• Tao, Analysis (I and II)

  Tao’s approach is a rigorous construction of analysis beginning with the foundations of the number system. Proofs are largely left as exercises, and it lacks much intuitive discussion, so I don’t think it’s appropriate for self-study except in combination with some other book. It’s published by Hundustan Book Agency and sold in the USA via AMS (AMS store page). It can be ordered from India via these links:

  • I: http://www.abebooks.com/products/isbn/9789380250649
  • II: http://www.abebooks.com/products/isbn/9789380250656

• Shilov, Elementary Real and Complex Analysis (Dover)

  This translation of a Russian book is an exception to what I said at the beginning of this section: it doesn’t cover the Lebesgue integral. Instead, it includes a chapter on analytic functions, which (in the US system) is more typically covered in a course on complex analysis. Shilov covers the Lebesgue integral in his graduate-level sequel: Shilov and Gurevich, Integral, Measure and Derivative.

• Bressoud’s “Radial Approch” books
  • Bressoud, A Radical Approach to Real Analysis (2e)
  • Bressoud, A Radical Approach to Lebesgue’s Theory of Integration (1e)

  I discuss both of these books by Bressoud together. Bressoud’s concept is to teach analysis by tracing the history of the mathematical concepts, and the telling the story of the problems and controversies that played out over the course of the 19th Century that gave shape to modern analysis. He states that his main purpose is to teach you the math, though, not to teach you the history of mathematics.

  Bressoud summarizes A Radical Approach to Real Analysis thus: “The book begins with Fourier’s introduction of trigonometric series and the problems they created for the mathematicians of the early nineteenth century. It follows Cauchy’s attempts to establish a firm foundation for calculus, and considers his failures as well as his successes. It culminates with Dirichlet’s proof of the validity of the Fourier series expansion and explores some of the counterintuitive results Riemann and Weierstrass were led to as a result of Dirichlet’s proof.”

  The second book is summarized thus: “The story begins with Riemann’s definition of the integral, a definition created so that he could understand how broadly one could define a function and yet have it be integrable. The reader then follows the efforts of many mathematicians who wrestled with the difficulties inherent in the Riemann integral, leading to the work in the late nineteenth and early twentieth centuries of Jordan, Borel, and Lebesgue, who finally broke with Riemann’s definition. Ushering in a new way of understanding integration, they opened the door to fresh and productive approaches to many of the previously intractable problems of analysis.”

• Johnsonbaugh and Pfaffenberger, Foundations of Mathematical Analysis (1e)

• Ash. Real Variables, with basic metric space topology (FREE ONLINE, Dover)

  An introduction to real analysis. Includes solutions to problems!

• Trench. Introduction to Real Analysis (FREE ONLINE)

• Carothers, Real Analysis (1e)

• Erdman. Companion to Real Analysis (FREE ONLINE)

  Free course notes covering a wide range of topics.

Graduate analysis

These books are generally considered graduate level, and cover measure theory. (Measure theory is generally the big thing that’s left out of undergraduate analysis courses.)

• Royden (3e)

  Royden is one of the most popular classic real analyis texts. One thing to be aware of is that a fourth edition was published with Patrick Fitzpatrick as co-author, but it hasn’t been well-received: many felt that the third edition was in no need of revision, and the fourth introduced new errors. Since it was popular, many copies of the 3rd edition (of 1988) are still available on the used market at a reasonable price. I’ve also seen used copies of the 2nd edition floating around at secondhand stores.

• Rudin, Real and Complex Analysis (3e intl, 3e intl @AbeBooks)

  A traditional heavyweight, also known as “Big Rudin”, “Adult Rudin” or “Daddy Rudin”. Extremely expensive, but you can buy the Indian edition.

• Kolmogorov and Fomin, Elements of the Theory of Functions and Functional Analysis (Martino, Dover)

  Kolmogorov was one of the great mathematicians of the 20th Century. Translated from Russian. This seems to be the preferred translation. (It’s roughly the same book as Silverman’s translation, which is titled Introductory Real Analysis.)

• Folland, Real Analysis: Modern Techniques and Their Applications (2e)

  “Encompassing several subjects that underlie much of modern analysis, the book focuses on measure and integration theory, point set topology, and the basics of functional analysis.”

  It’s said to be an excellent book, and it’s popular. Too bad it’s so expensive. There’s an older edition but even that one isn’t cheap.

• Bartle. The Elements of Integration and Lebesgue Measure (1e, 1e intl @AbeBooks)

  Well-regarded text on the more advanced topics of real analysis.

• Taylor, General Theory of Functions and Integration (Dover)

  Another classic text (1965). It is notably clear in its explanations. This one doesn’t seem to be very popular, but personally I like it.

• Bass, Real Analysis for Graduate Students (FREE ONLINE, 2e self-published paperback)

  Designed as an overview of all the real analysis that a grad student should need to pass a prelim in real analysis. Not intended to teach it to you the first time.

• Stein and Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces (1e)

  One of four books form a series (Princeton Lectures in Analysis) at the level of advanced undergraduate or beginning graduate analysis. They are widely praised for their quality, but they are also a bit pricey for what you get (and used copies aren’t much cheaper).

• Shilov and Gurevich, Integral, Measure and Derivative (Dover)

  Shilov’s graduate analysis textbook on Lebesgue integration and measure theory.

• Lieb and Loss, Analysis (2e)

  Largely focuses on applications. Includes the Fourier transform, Sobolev spaces and the calculus of variations. A more advanced book that seems to assume a prior introduction to measure theory.

• Folland, A Guide to Advanced Real Analysis (1e)

  “This book is an outline of the core material in the standard graduate-level real analysis course. …. Proofs, however, are generally presented only as sketches, in such a way that the key ideas are explained but the technical details are omitted.” (Usually when someone refers to the Folland book, though, it’s his other one at this level.)

Problem books

• Larson. Problem-Solving Through Problems (1e PB)

• Aksoy and Khamsi. A Problem Book in Real Analysis (1e)

  Covers basic topics in real analysis. Includes solutions!

• Erdman. A ProblemText in Advanced Calculus (FREE ONLINE)

• Moise. Introductory Problem Courses in Analysis and Topology (1e PB)

• Shakarchi. Problems and Solutions for Undergraduate Analysis (1e)
  • Shakarchi. Problems and Solutions for Complex Analysis (1e)

  All the exercises from Serge Lang’s Undergraduate Analysis and Complex Analysis respectively, with solutions. These are apparently intended for use with Lang’s textbooks, but since the problems are included I guess you could use them on their own as well. (Significant because they cost as much as the textbooks themselves.)

• Volkovyskii, Lunts, Aramanovich. A Collection of Problems on Complex Analysis

• Gelbaum. Problems in Real and Complex Analysis (1e HC)

• Kirillov and Gvishiani. Theorems and Problems in Functional Analysis (1e)

• Yeh. Problems and Proofs in Real Analysis: Theory of Measure and Integration (1e)

  A graduate-level set of problems and solutions.

• Furdui. Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis (1e)

• Aliprantis and Burkinshaw. Problems in Real Analysis (2e)

• Kaczor and Nowak. Problems in Mathematical Analysis (Vol I; Vol II; Vol III)

• Pólya and Szegö. Problems and Theorems in Analysis
  • Vol I: Series, Integral Calculus, Theory of Functions (Vol I)
  • Vol II: Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry (Vol II)
• Gasinski and Papageorgiou.
  • Exercises in Analysis: Part 1 (1e)
  • Exercises in Analysis: Part 2: Nonlinear Analysis (1e)

  A huge, brand-new collection of problems on analysis (each book is over 1000 pages). This looks to be the kind of thing you’re only interested in if you’ll be specializing in real analysis. “This nearly encyclopedic coverage of exercises in mathematical analysis is the first of its kind and is accessible to a wide readership. Graduate students will find the collection of problems valuable in preparation for their preliminary or qualifying exams as well as for testing their deeper understanding of the material.”

• Rădulescu, Rădulescu, Andreescu. Problems in Real Analysis: Advanced Calculus on the Real Axis (1e)

Fourier analysis

• Stein and Shakarchi, Fourier Analysis: An Introduction (1e)
• Hubbard, The World According to Wavelets (1e)
• Tolstov, Fourier Analysis (Dover)

Manifolds

• Munkres, Analysis On Manifolds (1e)
• Spivak, Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus (1e)
• Loomis and Sternberg. Advanced Calculus (WS Revised Edition)

See also: Differential Geometry.

Complex Analysis

• Bak and Newman, Complex Analysis (3e)

  This has a reputation as a relatively easy introduction to complex analysis, but it covers less material than others.

• Stein and Shakarchi, Complex Analysis (1e)

  A very well-liked intro to complex analysis, and perhaps the best-loved of the S&S series. Still expensive.

• Ahlfors, Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable (3e, 3e intl at AbeBooks, 3e intl, 2e)

  The traditional textbook for a complex analysis course. Typically considered old and dry these days, but still useful.

• Flanigan, Complex Variables: Harmonic and Analytic Functions (Dover)

• Conway, Functions of One Complex Variable (Vol I: 2e, 2e intl at AbeBooks; Vol II: 1e)

• Needham, Visual Complex Analysis (1e)

  An introduction to complex analysis in pictures.

• Wegert. Visual Complex Functions: An Introduction with Phase Portraits (1e)

  Another pictorial intro to complex analysis.

• Schwerdtfeger, Geometry of Complex Numbers (Dover)

• Gamelin. Complex Analysis (1e)

• Ash and Novinger, Complex Variables (FREE ONLINE, 2e Dover)

  Contain solutions. Author’s page: http://www.math.uiuc.edu/~r-ash/CV.html

• Markushevich. *Theory of Functions of a Complex Variable (AMS Chelsea revised English edition)

• Ablowitz and Fokas. Complex Variables: Introduction and Applications (2e)

Functional Analysis

• Stein and Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis (1e)
• Akhiezer and Glazman, Theory of Linear Operators in Hilbert Space (Dover)
• Griffel, Applied Functional Analysis (Dover)
• Shilov, Elementary Functional Analysis (Dover)
• Rudin, Functional Analysis (McGraw-Hill intl @AbeBooks)
• Halmos, Introduction to Hilbert Space and the Theory of Spectral Multiplicity (Martino)
• Halmos, A Hilbert Space Problem Book (2e)
• Erdman. Functional Analysis and Operator Algebras: An Introduction http://web.pdx.edu/~erdman/FAOA/functional_analysis_operator_algebras_pdf.pdf
• Aliprantis and Border. Infinite Dimensional Analysis: A Hitchhiker’s Guide (3e)

Dynamical systems

• Strogatz. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (2e, 1e)
• Holmgren. A First Course in Discrete Dynamical Systems (2e)
• Devaney. An Introduction to Chaotic Dynamical Systems (2e)
• Sternberg. Dynamical Systems (Dover)
• Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos (2e)
• Glendinning. Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations (2e)
• Hirsch, Smale, Devaney. Differential Equations, Dynamical Systems, and an Introduction to Chaos (3e)

Asymptotic methods, perturbation theory

• Bender and Orszag. Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory (1e)
• de Bruijn. Asymptotic Methods in Analysis (Dover)
• Simmonds and Mann. A First Look at Perturbation Theory (Dover 2e)
• Bellman. Perturbation Techniques in Mathematics, Engineering and Physics (Dover)

Numerical analysis

See Numerical Methods in the CS section.