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.
Best Answer
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