Real Analysis

by Frank Morgan

This book is written by award-winning author, Frank Morgan. It offers a simple and sophisticated point of view, reflecting Morgan's insightful teaching, lecturing, and writing style. Intended for undergraduates studying real analysis, this book builds the theory behind calculus directly from the basic concepts of real numbers, limits, and open and closed sets in $\mathbb{R}^n$. It gives the three characterizations of via epsilon-delta, sequences, and open sets. It gives the three characterizations of as "closed and bounded," via sequences, and via open covers. Topics include Fourier series, the Gamma function, metric spaces, and Ascoli's Theorem. This concise text not only provides efficient proofs, but also shows students how to derive them. The excellent exercises are accompanied by select solutions. Ideally suited as an undergraduate textbook, this complete book on real analysis will fit comfortably into one semester. Frank Morgan received the first Haimo Award for distinguished college teaching from the Mathematical Association of America. He has also garnered top teaching awards from Rice University (Houston, TX) and MIT (Cambridge, MA).

151 pages, Hardcover

First published January 1, 2005

Reviews

[Brett · Rating 3 out of 5]

You'd think Cauchy and other mathematicians wasted their lives away with the amount of space this book gives to some of their ideas. Indeed you wonder if someone could ever learn anything with such a short exposition on some topics. Infinity - 4 pages. Sequences - 6 pages. Functions and limits - 3 pages. Lebesgue Theory - 2 pages! Moreover treats each subject as a complete chapter, so you get some odd ones, like Chapter 6: Continuous Functions, pages 33-34. Terse is an overstatement. Morgan has a very unique idea about how to approach advanced calculus.

This was the book I used in my real analysis class, and at the time I was a little puzzled. Most of the time I had to use the class lectures to learn the material and I would use this book only as a reference. But as I have continued in my research I have found the book more useful. I don't need to search through a sea of explanatory text to get to the theorems I need. If I have a question about sequences of functions, I turn to pages 75-78, that's it. And in a way, that makes sense. Good mathematics doesn't need a lot of text if the theorems are laid out well. Far too often, however, certain problems come up and then we need help. This book seldom provides it.

[Daeus · Rating 4 out of 5]

Great and incredibly concise summary of real analysis. I appreciate real analysis concepts and their impact as the foundation of calculus (which can model change for engineering, etc), but theory is not really my thing (esp Topology, which feels like binary vs coding in an established language, or deep grammar principles vs writing). Still, this book is great overview/summary of key concepts - better as a refresher than a first introduction, and brilliantly simple. 

Quotes
- "We like sequences to converge, but most don't. Fortunately, most sequences have subsequences which converge."
- "The third of the big Cs requisite for calculus, after 'continuous' and 'compact,' is 'connected."
- "The Mean Value Theorem says that there is a point c where the instantaneous slope equals the average slope from a to b." (Image on pg 62). "The following theorem is the only reason you need the Mean Value Theorem to do calculus.... "Corollary of the MVT. On an interval where f' is always 0, f is constant."
- "The Fundamental Theorem of Calculus is most popular for its second part, which says that you can integrate just by antidifferentiating, instead of doing painful limits of Reimann sums. Both parts essentially say that integration and differentiation are opposites. It is quite remarkable that there is any relationship between integration and differentiation, between area and slope, between Reimann sums and limits of ratios of change."
- "The remarkable underlying mathematical fact is that every smooth function on [-pi, pi] can be decomposed as an infinite series in terms of sines and cosines, called Fourier series. This is in strong contrast to the fact that only real-analytic functions are given by Taylor series in powers of x. Apparently, for decomposition, sines and cosines work much better than power of x."
- ..."define e, perhaps as lim(1+1/n)^n." ... "e =... sum(1/n!) = 2.71828...."
- "Fortunately, there is a nice extension of (x-1)! from integers to real numbers greater than 1, called the Gamma Function." Just noting that for myself since gamma functions sometimes come up in statistics via the gamma distribution.
- "To solve problems, one needs compactness in order to extract from a sequence of approximate solutions and exact solution in the limit. For sophisticated problems, the desired solution is not just a number, but rather a function, perhaps describing an economically ideal schedule of production or the most efficient shape of an airplane wing."

[Farrokh · Rating 3 out of 5]

This book contains most of the important part of the analysis mathematic and you can review all you knowledge. It has 4 parts.
First part is Real Number and limits which is about the pre-calculus subject like definition of limitation, the other part is topology it contains some definition of set and subsets in analysis mathematic, next part is calculus it just has some definition and some good exercises and problems the last part is Metric Spaces. This part is really short and includes some view that what is the real analysis mathematic.