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What books on analysis would people recommend after someone has finished all three by Rudin (Principles of Mathematical Analysis, Real and Complex Analysis, and Functional Analysis)?

I am looking for well-organised books which go deep: either 1-2 which are broad in scope, or, if no single book at this advanced level offers a lot of breadth, then a set which pack a lot of breadth when considered together.

Yiorgos S. Smyrlis
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Harry Macpherson
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5 Answers5

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As others have said, it really depends on what your interests are. There are dozens of directions you could go, each of which can consume more than a lifetime of work and study. I don't know what your situation is, but at some point one leaves the textbooks behind (mostly, at least) and begin to focus on research papers (both contemporary and older), as very little of the enormous research literature actually makes it into monographs and treatises, to say nothing of textbooks. For example, almost none of the various results dealt with in the references I gave yesterday in my answer to $\alpha$-derivative (concept) can be found in any books. (The only thing I can think of off-hand is the Auerbach/Banach paper, whose results I believe can be found in Eduard Cech's 1969 text Point sets.) Of course, much of this research literature consists of tangled paths probably few would want to follow anyway . . .

That said, if for whatever reason you wish to devote two or three years going through a textbook/monograph, I suggest one of the following two:

Nelson Dunford and Jacob Schwartz's multi-volume series Linear Operators

Zygmund's Trigonometric Series

Each of these is a classic and each contains a huge amount of mathematics. Probably the Dunford/Schwartz series is the better fit for you, I suspect, as it has a large number of carefully thought out exercises.

Depending on your interests, however, any of the following should also work, along with dozens of other paths I (or others) could easily come up with.

Pertti Mattila's Geometry of Sets and Measures in Euclidean Spaces, perhaps followed by Herbert Federer's Geometric Measure Theory

Lindenstrauss/Tzafriri's 2-volume work Classical Banach Spaces, perhaps followed by Lindenstrauss/Preiss/Tišer's Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces

Gilbarg/Trudinger's Elliptic Partial Differential Equations of Second Order, perhaps followed by Heinonen/Kilpeläinen/Martio's Nonlinear Potential Theory of Degenerate Elliptic Equations

Steven Krantz's Function Theory of Several Complex Variables

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Dave L. Renfro
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    (+1) For the nice list but for the benefit of the OP: Do not go and try to work through Gilbarg/Trudinger. Just don't. – J.R. Feb 05 '14 at 21:11
  • @TooOldForMath: I actually took a graduate course that used their text, but we didn't get very far into it. My recollection is that it was like reading a difficult novel in a language you don't know with the aid of a dictionary, but I figured this was because most of the class consisted of 2nd year M.A. students (this being, at the time, a non-Ph.D. granting university) at roughly the level of having survived baby Rudin level analysis. I knew rather more than this, but not enough of the right things. – Dave L. Renfro Feb 05 '14 at 21:27
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    Haha, I also took such a course, but the professor switched to using his own notes eventually.. it's just not a good text book, more of a reference source. – J.R. Feb 05 '14 at 21:33
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    Surely Hörmander's SCV text deserves a mention. –  Feb 06 '14 at 04:25
  • @Mike: To be honest, I wasn't sure what the analog of Dunford/Schwartz or Zygmund (or Federer) was for several complex variables. However, I do know that if Gilbarg/Trudinger doesn't have the breadth Harry Macpherson is looking for, then he should look at Hörmander's 4-volume treatise The Analysis of Linear Partial Differential Operators! Nonetheless, I believe Dunford/Schwartz (at least Volume 1) would be a good place for him to go next. Among other things, it has a very nice (but extremely rapid) summary of many classically important prerequisites in the first few chapters. – Dave L. Renfro Feb 06 '14 at 15:17
  • Those not familar with Dunford/Schwartz's reputation might find it useful to pursue this google search: Dunford Schwartz problems worked – Dave L. Renfro Feb 06 '14 at 15:18
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Direction I. Harmonic Analysis: Get the books of L. Grafakos (Classic and Modern Harmonic Analysis).

Direction II. PDEs: Start with the book of Evans (PDEs), and get also Adams's Sobolev Spaces.

Direction III. Operators. Get the books of Reed and Simon, also T. Kato's Perturbation Theory. Also the book of P. Lax is a nice reading.

Direction IV. Complex Analysis. Get Foster's Riemann surfaces and Narashiman's Several Complex Variables.

Yiorgos S. Smyrlis
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Real Analysis by Serge Lang and maybe Dieudonne: Foundations of Modern Analysis.

Martín-Blas Pérez Pinilla
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Well I would suggest you visit Terence Tao's blog or website. They will definitely offer tons of material and problems to think.

Sandeep Thilakan
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how about Duistermaat: Multivariable Real Analysis (a two volume set)?

also, Szego & Polya: Problems in Analysis

hilbert53
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