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It seems that nearly every graduate program in the U.S. requires an extensive education in Analysis clearly demonstrating its importance not only as a subfield itself, but as a foundation for other areas of abstract mathematics.

I am interested in doing research in Analysis as a graduate student. I am curious how long it takes a student to get to the level of serious research in the subject.

As subjective as this question is, I am curious to know from which texts some of you learned Analysis i.e. from Rudin (the family), Royden, Pugh, Bartle, Tao, Torchinsky, etc.

How many classes did it take to be near or at the level of research in the area? Which texts helped you get there?

Also, do you know of any summer programs in Analysis at your institution undergraduates can enroll in?

Thanks in advance!

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closed as primarily opinion-based by Matthew Towers, Jyrki Lahtonen, Henrik, Cesareo, mvw Jun 23 '18 at 18:30

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ I think a variety of books provides different emphases & perspectives. While I am not a mathematician, books that I found helpful were Functional analysis by Kantorovich & Akilov, Introductory real analysis by Kolmogorov & Fomin. At a more elementary level, I enjoyed Marsden's Elementary Classical Analysis. Rudin is an amazing text, but I found it a bit dense from an introductory perspective. Royden, Spivak, etc, were also helpful. $\endgroup$ – copper.hat Jun 23 '18 at 7:07
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In my undergrad, I took two one-semester courses following Royden's Real Analysis. I liked this book because Royden (generally) has a sufficient amount of detail. I found it a good grounding in the fundamentals of measure theory and metric spaces.

After this, I took a course in Fourier analysis that used many textbooks (among which my favourite was Katznelson's An Introduction to Harmonic Analysis) and a course in functional analysis that used Macluer's Elementary Functional Analysis (an introductory text but very well presented - I imagine very appropriate for self-study).

It was during this time (the last year of my undergrad) that my advisor lent me her copy of Rudin's Real and Complex Analysis. This changed my outlook on analysis. Rudin writes in a way that I find simultaneously enlightening and challenging. The overall effect is stimulating. I feel like this book was important in training as an analyst and still exerts an influence on the way I think.

I found that these texts were sufficient for me to acquire strong foundations. In fact, these four texts are among those that can be found at my desk for quick access. For getting to research-level analysis, my experience is that each area has its own set of knowledge and techniques at people's fingertips. Reading specialised textbooks and well written papers in a particular area is, in my opinion, the best way to approach research. An advisor or other specialist can be a great source of suggestions to find these things.

For undergraduate summer programmes, definitely speak to people in your department (repeating @Thomas, above).

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People will have different experiences of doing research.

Typically in a graduate program in the US one starts with taking general classes. After covering some basic classes (often in algebra, topology, and analysis (and maybe others)) you might start to take more specialized classes. At this point you might also start working with a specific professor on a project. It is really dependent on the project and you how "fast" you get to do actual research. Some professors will have students start almost immediately with smaller projects. Others will insist that you cover certain background before giving you a project to work on.

So, it is really difficult to say how long to "get to a serious level". My opinion is that it all is serious. Doing research now I often find my self consulting graduate (or even undergraduate) level textbooks. Often I have questions that might be answered in a graduate level class! In general I would say to not worry so much about whether the level is "serious".

As for books, I used both Rudin and Royden.

The best way to know about summer programs is to ask faculty in the math department. They often have an undergraduate advisor. You can also ask your current instructors if they know anything. At least you would probably need to seek our letters of recommendation from past instructors. You institution might even have funding for you. You can also look online for programs. I always get excited when a student approaches me about wanting to do a summer program!

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As far as textbooks are concerned, I learned Real Analysis mainly from Michael Spivak's Calculus. My contact with Walter Rudin's textbooks started only after graduation, but I also have a very high opinion about them.

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In what follows I’ve restricted myself to those books I actually used at the time (nothing from after about 1991 or 1992), so this does not include anything that appeared afterwards.

I had a one-semester course out of Royden (covered most of the book --- it was a fairly fast-paced course), a one-semester course out of Taylor’s General Theory of Functions and Integration (covered the middle third of the book, this being at a different university), and a two-semester course using Wheeden/Zygmund's Measure and Integral that was taught by Torchinsky (6 years before his 1988 book appeared, but much of what's in his book was included, in particular all the stuff on cardinal and ordinal numbers and order types). But these are introductory texts that Ph.D. qualifying exams are based on, and of course I and pretty much everyone else was also familiar with several other books, in my case probably the most significant were (in order of how much I studied the book) Measure Theory by Halmos, Real and Abstract Analysis by Hewitt/Stromberg, Measure Theory by Cohn, and Integration by McShane. Also, Real and Complex Analysis by Rudin (first third of the book) was often recommended, and many students used it for Ph.D. qualifying exam preparation, but for whatever reason I never looked at it all that much, and in fact it was only 4 or 5 years ago that I finally wound up getting a copy of Rudin’s book (saw it at a used bookstore).

For what it’s worth, a couple of books that I recall being recommended very often were (this being late 1970s to early 1980s) Functional Analysis by Riesz/Sz.-Nagy (first third of the book, the three chapters covering integration) and Linear Operators. Part I. General Theory by Dunford/Schwartz (also first third of the book, which is also the three chapters covering integration), but I never wound up looking at them very much. However, in looking at these two books right now, I believe the first third of Riesz/Sz.-Nagy would make for a good summer project if someone wants a refresher in classical analysis topics. But not Dunford/Schwartz, unless you really want to dig deep into set function machinery stuff. Another book (2 volumes, actually) that was often suggested is Theory of Functions of a Real Variable by Natanson, which incidentally I later wound up using often as a reference, but because everything is restricted to the real line the drawback is that you’re not going to see any general measure theory.

What you should read AFTER the basic first year course material will depend hugely on what area of analysis you intend to work in. In my particular case, and this is a rather outlying area (but which I find fascinating), the books that I wound up making the most use of include Real Functions by Goffman, A Primer of Real Functions by Boas (1981 edition), A Second Course on Real Functions by van Rooij/Schikhof, Measure and Category by Oxtoby, The Geometry of Fractal Sets by Falconer, Fractal Geometry by Falconer, Differentiation of Real Functions by Bruckner (1978 edition), Real Functions by Thomson, and Theory of the Integral by Saks.

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  • $\begingroup$ Thank you for the comprehensive answer and great references! I'll check those out. $\endgroup$ – coreyman317 Jun 23 '18 at 1:05
  • $\begingroup$ Keep in mind my comment "and this is a rather outlying area" regarding the references in the last paragraph, and regarding the other references, remember that nothing after about 1990 or 1991 is included. $\endgroup$ – Dave L. Renfro Jun 23 '18 at 8:08

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