What books on analysis after someone has finished all 3 by Rudin? 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.
 A: 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
A: 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.
A: Well I would suggest you visit Terence Tao's blog or website. They will definitely offer tons of material and problems to think.
A: Real Analysis by Serge Lang and maybe Dieudonne: Foundations of Modern Analysis.
A: how about Duistermaat: Multivariable Real Analysis (a two volume set)?
also, Szego & Polya: Problems in Analysis
