**220 of 238 people found the following review ... energetic**

In the post below, I link to the text Offspring #1 will apparently be using in her Real Analysis class. Just to see if it would help me grasp what on earth "real analysis" might mean, I read the Amazon reviews, and found this one. Amusing reading, even if it does leave me no closer to understanding the topic than before, and pitifully glad to have been a liberal arts major.

OK... Deep breaths everybody...(Spelling fixed. Hey, he's not an English major, is he?)

It is not possible to overstate how good this book is. I tried to give it uncountably many stars but they only have five. Five is an insult. I'm sorry Dr. Rudin...

This book is a good reference but let me tell you what it's really good for. You have taken all the lower division courses. You have taken that "transition to proof writing" class in number theory, or linear algebra, or logic, or discrete math, or whatever they do at your institution of higher learning. You can tell a contrapositive from a proof by contradiction. You can explain to your grandma why there are more real numbers than rationals. Now it's time to get serious.

Get this book. Start at page one. Read until you come to the word Theorem. Do not read the proof. Prove it yourself. Or at least try. If you get stuck read a line or two until you see what to do.

Thrust, repeat.

If you make it through the first six or seven chapters like this then there shall be no power in the verse that can stop you. Enjoy graduate school. You're half way there.

Now some people complain about this book being too hard. Don't listen to them. They are just trying to pull you down and keep you from your true destiny. They are the same people who try to sell you TVs and lobotomies.

"The material is not motivated." Not motivated? Judas just stick a dagger in my heart. This material needs no motivation. Just do it. Faith will come. He's teaching you analysis. Not selling you a used car. By the time you are ready to read this book you should not need motivation from the author as to why you need to know analysis. You should just feel a burning in your chest that can only be quenched by arguments involving an arbitrary sequence {x_n} that converges to x in X.

Finally, some people complain about the level of abstraction, which let me just say is not that high. If you want to see abstraction grab a copy of Spanier's 'Algebraic Topology' and stare at it for about an hour. Then open 'Baby Rudin' up again. I promise you the feeling you get when you sit in a hottub for like twenty minutes and then jump back in the pool. Invigorating.

No but really. Anyone who passes you an analysis book that does not say the words metric space, and have the chaptor on topology before the chaptor on limits is doing you no favors. You need to know what compactness is when you get out of an analysis course. And it's lunacy to start talking about differentiation without it. It's possible, sure, but it's a waste of time and energy. To say a continuous function is one where the inverse image of open sets is open is way cooler than that epsilon delta stuff. Then you prove the epsilon delta thing as a theorem. Hows that for motivation?

Anyway, if this review comes off as combative that's because it is. It's unethical to use another text for an undergraduate real analysis class. It insults and short changes the students. Sure it was OK before Rudin wrote the thing, but now? Why spit on your luck? And if you're a student and find the book too hard? Try harder. That's the point. If you did not crave intellectual work why are you sitting in an analysis course? Dig in. It will make you a better person. Trust me.

Or you could just change your major back to engineering. It's more money and the books always have lots of nice pictures.

In conclusion: Thank you Dr. Rudin for your wonderfull book on analysis. You made a man of me.

## 5 Comments:

Real analysis is the study of real numbers, as opposed to rational or complex numbers. The basic topics of real analysis are the same four basic topics found in freshman calculus: limits, continuity, differentiation, and integration. But real analysis is far more rigorous in its approach to these topics than calculus courses generally are. For example, the theorems of real analysis always assume multidimensional space rather than the nice two-or-three dimensional space that is usually referenced in basic calculus, and in real analysis the difference between "countably infinite" and "uncountably infinite" is crucial - a distinction that is usually ignored in freshman calculus. Indeed, I think of real analysis as the class that makes rigorous all those messy details that freshman calculus instructors tend to gloss over so as to avoid confusing their students. Also, because real analysis is usually taught in the first semester of graduate school, it is the course where professors (and textbook writers) tend to get extremely fussy about proofs. Rigor is required to a degree that most real analysis students have not experienced before.

In a nutshell: seriously advanced math for anyone other than graduate students in a highly technical field. If your 13-year-old is the kind of girl who taught herself calculus in her spare time, aced differential equations without breaking a sweat, and thought that abstract algebra was "the first interesting math class I've ever had," then I would say that real analysis is the next logical step. If not, you might want to wait on this until she's well into college. I'm just saying.

Joel

Joel,

Thanks for that explanation. Unfortunately I wasn't far through the first paragraph when my brain started to make the fluttery "la la la" I remember so well from Mr. Hill's calculus class. I did take the Calculus AP, and got a 3, which was very exciting to me because it meant I could place out of math at Big State U. and never, ever have to take a math course again.

Offspring #1, who didn't inherit my math genes, did finish calculus but I don't know about those other courses you mentioned; it's been some time since I recognized any of the math she does. She's been taking math courses at Big State U. for a while now--not for credit (she's too young) but by consent of instructor--and plans to do the same with Real Analysis. Her last course, Number Theory (again, I know the words, but I don't know what they signify) seemed to involve lots of proof-writing (shudder), so I suppose she's good to go in that area. But I'm only in charge of the arts and sciences around here, so what do I know.

Your 13-year-old took number theory? Goodness, maybe she will be OK in real analysis.

Kids these days . . . .

Joel

Unfortunately I wasn't far through the first paragraph when my brain started to make the fluttery "la la la"Whoa, that was exactly my reaction too. If my children ever turn out to have the mathematical genius of #1 (not likely) I'm turning over the advanced formation of their tender minds to someone far more qualified in that regard.

I love that review!

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