Counterexamples in analysis
These counterexamples deal mostly with the part of analysis known as "real variables." The 1st half of the book discusses the real number system, functions and limits, differentiation, Riemann integration, sequences, infinite series, more. The 2nd half examines functions of 2 variables, plane sets, area, metric and topological spaces, and function spaces. 1962 edition. Includes 12 figures.
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- A favorite on my bookshelf:
Bernard R.\ Gelbaum and John M.\ H.\ Olmsted, {\it Counterexamples in Analysis,\/} Holden-Day, San Francisco, 1964.\ \
- Gelbaum-Olmsted Counterexamples in analysis is a standard reference for various pathological examples in analysis:
- I agree and I don't know why the myth persists that [baby] Rudin is the best way to learn analysis. It's certainly not so for a lot of people who could learn analysis perfectly well from friendlier (but not necessarily any less rigorous) sources. Rudin requires a lot of self-motivation.
In addition to Abbott which you mentioned,
"How to Think About Analysis" (Alcock), good gentle introduction for anyone who tries to dive into an analysis textbook and hits concrete.
"Real Mathematical Analysis" (Pugh), good, rigorous approach that doesn't require quite as much effort to power through as Rudin does, although some of the exercises are tough.
"Analysis I" and II (Tao), is good and starts from scratch building numbers from the Peano axioms, so there are no other unproven assumptions underlying everything later.
"Counterexamples in Analysis" (Gelbalm & Olmstead), very helpful for understanding some pathological cases that break theorems one might intuitively believe are true.
There's no reason today why someone can't grab pdfs of all of the above (even Rudin, although it's never been digitally typeset), from any friendly internet library, and use them all to build a better understanding rather than relying on one alone.
- > And guess what, you even need complex numbers.
That's not true, and that is the point. To prove the main theorems of calculus, you don't need complex numbers, but you do need irrationals.
When you first learn calculus, you do one or two epsilon-delta proofs, and then your teacher gets a little hand wavy about limits and you move on to the real work of derivatives and integrals, cause the limits stuff intuitively makes sense. When you continue on in Real Analysis or Topology of the Real Line, you discover that your intuition lied to you, and concepts like open and closed sets and intersections and accumulation points are important and are in general non-obvious.
Counterexamples In Analysis
https://www.amazon.com/Counterexamples-Analysis-Dover-Books-...
(Pdf version) https://pdfs.semanticscholar.org/a4e7/eb352e4c44bf75d8fabaf7...
- By the time you can grok the lesbegue measure?
Edit: A great way to unlearn bad intuition is through the study of counterexamples. A good starting point might be chapter 1 of Counterexamples in Analysis [1].
[1]https://pdfs.semanticscholar.org/a4e7/eb352e4c44bf75d8fabaf7...
- Evar D. Nering, Linear Algebra and Matrix Linear Algebra and Matrix Theory*
Kenneth Hoffmann And Ray Kunze. Linear Algebra, 2nd Edition, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.
https://www.zuj.edu.jo/download/linear-algebra-2nd-edition-k...
Halmos, Finite Dimensional Vector Spaces
George E. Forsythe and Cleve B. Moler, Computer Solution of Linear Algebraic Systems
Paul R. Halmos, Naive Set Theory, Van Nostrand, Princeton, NJ, 1960.
More has been done since this book, but this book is a gorgeous introduction to axiomatic set theory. So, even people who want to dig into the latest work would do well to have this as the first book. And for people wanting to read any of the more advanced material here, knowledge of this book will be from good to have to important.
https://piazza-resources.s3.amazonaws.com/is25bi6c6o1oh/isv1...
Walter Rudin, Principles of Mathematical Analysis
The third edition is a lot better than the first two.
H. L. Royden, Real Analysis: Second Edition
Beautifully written, elegant, but maybe don't work way too hard on the exercises about upper/lower semi-continuity, and there is a better summary than Littlewood's three principles.
Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis
John C. Oxtoby, Measure and Category: A Survey of the Analogies between Topological and Measure Spaces
Walter Rudin, Real and Complex Analysis
Walter Rudin, Functional Analysis
Leo Breiman, Probability
Kai Lai Chung, A Course in Probability Theory, Second Edition
Jacques Neveu, Mathematical Foundations of the Calculus of Probability
Erhan Cinlar, Introduction to Stochastic Processes
J. L. Doob, Stochastic Processes
I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes I, II
Donald E. Knuth, The TeX book
Donald E. Knuth, The Art of Computer Programming, Second Edition
Leo Breiman, "Statistical Modeling: The Two Cultures," Statistical Science, Vol. 16, No. 3, 199–231, 2001.
Paul R. Halmos, "The Theory of Unbiased Estimation", Annals of Mathematical Statistics, Volume 17, Number 1, pages 34-43, 1946.
Paul R. Halmos and L. J. Savage, "Application of the Radon-Nikodym Theorem to the Theory of Sufficient Statistics", The Annals of Mathematical Statistics, Volume 20, Number 2 (1949), 225-241.
https://projecteuclid.org/download/pdf_1/euclid.aoms/1177730...
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