# Textbook Review – “A First Course in Real Analysis”

This isn’t a subject about analyzing data, it’s about building calculus out of the properties of the real numbers. It’s a field that is likely very interesting to anyone interested in math for the sake of math.

### General Information

Title: A First Course in Real Analysis (Second Edition)
Authors: M.H. Protter and C.B. Morrey
Original Publication Date: 1977 (first edition), 1991 (second edition)
ISBN: 0-387-97437-7 and 3-540-97437-7 are both listed on the cover.
Cover Price: None; I paid just over \$100 Canadian for the hardcover.
Publisher: Springer-Verlag
Buy from: Amazon.com or Amazon.ca

### Subject Matter

This deals with real analysis, the field of math that studies properties of the real number systems and calculus. The text begins by determining the smallest number of axioms required to uniquely define the real number line, and then continues on by deriving calculus based on these axioms. The chapter titles are:

1. The Real Number System
2. Continuity and Limits
3. Basic Properties of Functions on R1
4. Elementary Theory of Differentiation
5. Elementary Theory of Integration
6. Elementary Theory of Metric Spaces
7. Differentiation in RN
8. Integration in RN
9. Infinite Sequences and Infinite Series
10. Fourier Series
11. Functions Defined by Integrals; Improper Integrals
12. The Riemann-Steiltjes Integral and Functions of Bounded Variation
13. Contraction Mapping, Newton’s Method, and Differential Equations
14. Implicit Function Theorems and Lagrange Multipliers
15. Functions on Metric Spaces; Approximation
16. Vector Field Theory; the Theorems of Green and Stokes

This also has appendices titled “Absolute Value,” “Solutions to Algebraic Inequalities,” “Expansions of Real Numbers in Any Base,” and “Vectors in EN.” The “EN” there should have N as a subscript. Similarly, the “R1” and “RN” above should have the 1 and N as superscripts with R in the “blackboard bold” font, since they are the 1 and N-dimensional real numbers respectively. They are, of course, represented properly in the textbook. I should also mention that I haven’t looked closely at chapters 10, 11, and 16, which were not used for the Analysis classes I used this for.

### High Point

The clear but methodical text. Everything has a proof along with it, unless the proof has been left up to the reader. (Most of those proofs come with hints, though.)

### Low Point

The discussion of metric spaces should have come sooner, since earlier chapters assumed certain results that don’t get proven until then.

### The Scores

I found the clarity of the text to be excellent. If this is the first formal math text you’re exposed to, you might disagree. However, once you’re used to the “Theorem, Proof, Lemma, Proof…” style, you’ll have no trouble moving through the material. I give it 6 out of 6.

The structure is excellent. Apart from the one sequencing issue I mentioned above, I had no troubles getting through the text. The formal style made this virtually a guarantee, though. The sequencing problem I had was a bit inconvenient, but when they assume something unproven, they include a pointer to the section of the text where it is proven, which can serve as an opportunity to go and read that section. It still makes sense that early, so even that is more of an inconvenience than a problem. I give it 5 out of 6.

The examples were sometimes limited, but most computational theorems had at least one example each. (When a theorem is only really used to prove another theorem, there’s no need to provide an example, is there?) Still, in some cases (like the one for integration methods of Fourier series solutions) more examples might have been helpful. I give it 4 out of 6.

The exercises were appropriate, plentiful, and the odd exercises come with answers (though not solutions.) Solutions to some would have been nice, so that they might serve as alternative examples, or so that the reader can really catch where mistakes are occurring. I give it 5 out of 6.

This is a very complete text. Of the four real analysis texts I have in my personal library, this is by far the most comprehensive. (Two don’t even cover functions with domain or range in more than one dimension.) This text can replace my existing calculus texts, and a fair portion of my differential equations texts, too. I give it 6 out of 6.

The editing was excellent. I didn’t notice any typos or other mistakes, additions and changes to this edition were seamless, and the grammatical structure and organization were excellent. The only things I’d change were structure and organization issues, not editing issues per se. I give it 6 out of 6.

Overall, this is an excellent textbook for the subject matter. It’s easy to learn from, and covers everything you’ll need for the real numbers and real number calculus. I give it 5 out of 6.

In total, A First Course in Real Analysis receives 37 out of 42.