# Learning real analysis

#### Pravit

My math finance program, like many others, suggests prospective students to have general knowledge of real analysis. I never had the chance to take analysis before graduating, but I do have three months this summer before my program starts. Would you suggest studying analysis or just brushing up on other stuff like linear algebra and probability? Are there some things in an MFE program that build off of analysis, or is it just there as a general prerequisite to screen out people who haven't done more advanced math? If the former, are there any topics I should focus on in particular?

Also, what books would you recommend for someone learning analysis by himself? I've seen "Yet Another Introduction to Analysis" mentioned elsewhere on this site, any other recs?

#### Bastian Gross

##### German Mathquant
Real Analysis in MFE

Hallo Pravit,

real analysis is quite different to linear algebra or probability stuff.
Nevertheless there are some things that build of analysis topics, like optimization, miscellaneous series expansions and so on.
But also real analysis is only a small fraction of math stuff in MFE.

So I would recommend to read a book by Walter Rudin:
Real and Complex Analysis

#### Sanket Patel

##### i do stuff
Victor Bryant - "Yet Another Introduction to Analysis" - has a very informal chatty style of writing while doing justice to the necessary rigor of Analysis.

Rudin's book, from I hear, is somewhat of a standard. But he can be rather dry and technical at times.

#### dstefan

##### Baruch MFE Director
So I would recommend to read a book by Walter Rudin:
Real and Complex Analysis

Rudin's book is what I used when learning real analysis myself. I think it is rather difficult to study by yourself by using Rudin's book, unless you have experience with abstract thinking for mathematical proofs. I looked once at the Measure, Integral and Probability book by Capinski and Kopp and it seemed OK.

So I would recommend to read a book by Walter Rudin:
Real and Complex Analysis

This book is meant for grad students in math who've already taken an undergrad course in real analysis. I don't recommend it for MFE students. The book you might have meant is Rudin's "Principles of Mathematical Analysis," informally known as "baby Rudin," which is widely used in undergrad analysis courses.

There's a plethora of excellent analysis books available today. Bryant is a good choice (as already pointed out by Patel). Another good book, for a concise overview, is Garrity's All the Mathematics You Missed: But Need to Know for Graduate School. The book I personally like best is Bressoud's A Radical Approach to Real Analysis, because of its masterful motivation and limpid pedagogy.

Keep in mind the key concepts in undergrad real analysis: a rigorous development of the properties of real numbers (particularly completeness, which distinguishes the reals from the rationals); the convergence of Cauchy sequences; and the use of limits and convergence to define and establish key results in continuity, differentiation, integration, and power series. Real analysis at this level gets its coherence and unity from the rigorous definition of limits and convergence (using epsilons and deltas) and their application to the areas given above. At a more advanced level, ideas from topology and measure theory are used, and this is the terminology used in rigorous probability (which can be found in the book by Capinski and Kopp, mentioned by dstefan)

#### chrisd

Both of Rudin's books are very good - particularly Principles of Mathematical Analysis, although be prepared to spend some time on it.

You might want to look into "Real Mathematical Analysis" by Charles Pugh, it's at the same level as Rudin in terms of rigour, covers the same material, but reads better and has some pictures. The book has a ton of exercises, ranging from moderate to very challenging (ie, harder than anything in Rudin, which is saying something). For what it's worth, it's also a good deal cheaper than Rudin's book.

#### Pravit

Wow, thanks everyone for all the excellent answers. I haven't really done any "real" math before; as an EE most of my math courses in college were computational and I didn't do many proofs, so I'm not sure if I'm ready to tackle Rudin given what I've heard about it.

The Measure, Integral, and Probability book Dr. Stefanica mentioned looks really good - my math finance program does mention specifically that students should have "general knowledge of real analysis, measure and integral, probability spaces", so maybe this is something I should look into. It even seems to cover some financial topics.

Based on what bigbadwolf said, I'm wondering if I should try to cover more basic topics in analysis before doing that book. The "All the math you missed" book does look like it could be very helpful.

May I recommend a wonderful textbook on Real Analysis, "Elements of Theory of Functions and Functional Analysis" by Kolmogorov and Fomin. It covers the material from their earlier edition, called "Introduction to Real Analysis" + some new topics, as the book was renewed in 1999 (after more than 20 years!).

I haven't really done any "real" math before; as an EE most of my math courses in college were computational and I didn't do many proofs, so I'm not sure if I'm ready to tackle Rudin given what I've heard about it.

... I'm wondering if I should try to cover more basic topics in analysis before doing that book.

One big hurdle --maybe the biggest? -- non-math types face is understanding the need for proofs: after all, the subject matter of intro real analysis seems to be the same as calculus: derivatives, integrals, and series; why then do we need to prove results that are intuitively or geometrically obvious (such as the intermediate value theorem or mean-value theorem)? The reasons are grounded in history: our intuition often leads us astray. There are counterexamples for results we may consider obvious; e.g. it might be intuitively clear that every continuous function should be differentiable everywhere except maybe at a finite number of points. That there may be continuous functions differentiable nowhere is a counterexample to our intuition. So we need a surer foundation for our knowledge and this is where careful proofs based on an axiomatic development of the real numbers come in handy (we hope). In more advanced analysis courses we learn that among continuous functions the set of differentiable functions is actually of measure zero; in other words, almost all continuous functions are non-differentiable. Results of this sort do find application: e.g., Brownian motion is continuous everywhere but differentiable nowhere. So rigorous definitions and careful proofs not only provide a sure(er) foundation for our understanding but also may be useful in other areas.

If I had to suggest a book for understanding proofs in general, I'd recommend Rotman's Journey into Mathematics: An Introduction to Proofs. It's going for a paltry $10 so you have no excuse to not get hold of a copy. (There are other books that also introduce the idea of proofs, but they're more expensive.) #### Sanket Patel ##### i do stuff One big hurdle --maybe the biggest? -- non-math types face is understanding the need for proofs: after all, the subject matter of intro real analysis seems to be the same as calculus: derivatives, integrals, and series; why then do we need to prove results that are intuitively or geometrically obvious (such as the intermediate value theorem or mean-value theorem)? The reasons are grounded in history: our intuition often leads us astray. There are counterexamples for results we may consider obvious; e.g. it might be intuitively clear that every continuous function should be differentiable everywhere except maybe at a finite number of points. That there may be continuous functions differentiable nowhere is a counterexample to our intuition. So we need a surer foundation for our knowledge and this is where careful proofs based on an axiomatic development of the real numbers come in handy (we hope). In more advanced analysis courses we learn that among continuous functions the set of differentiable functions is actually of measure zero; in other words, almost all continuous functions are non-differentiable. Results of this sort do find application: e.g., Brownian motion is continuous everywhere but differentiable nowhere. So rigorous definitions and careful proofs not only provide a sure(er) foundation for our understanding but also may be useful in other areas. If I had to suggest a book for understanding proofs in general, I'd recommend Rotman's Journey into Mathematics: An Introduction to Proofs. It's going for a paltry$10 so you have no excuse to not get hold of a copy. (There are other books that also introduce the idea of proofs, but they're more expensive.)

That was very well stated. =D

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