Great book--see my review. Uses the super-elegant Daniell-Stone approach for existence, but proceeds to develop integration as a tool, showing its breadth and versatility across a vast range of applications. (E.g., Halmos' showcase application--Haar measure and integration on locally compact groups--is one chapter here.)

Clear lectures on linear functional analysis with rigorous, nonrelativistic quantum mechanics (i.e., self-adjointness, spectral analysis, and scattering properties of the Schroedinger operator) and the beginnings of quantum fields as goals by Vol 4. The extensive annotated bibliography is especially welcome. The authors are likewise leading lights

Forget the text--just do all of the exercises, and you'll know the field cold. Be warned: It takes awhile. ("Linear Topological Spaces," by K and Namioka does the same thing for functional analysis but doesn't pull it off quite as well.)

Cliff Notes for PDEs! A good beginning and great review, but be sure you don't leave just as the party's getting good.

Strauss seems to have replaced this book as the standard undergraduate text. (To my mind it's an improvement, if only due to greater breadth, but mileage may well vary.) Many of us learned to love PDEs in this book, since it's been around a long time (1st-ed appeared in 1941). Also good, if you're stuck and ready to move beyond Farlow.

The reigning graduate-level text.

Strong emphasis on elliptic equations, but as good as it gets within its somewhat limited scope.

NYU's Courant Institute of Mathematical Sciences has historically been one of the world's foremost centers of research in PDEs. John was one of the first-generation of Courant greats after the War, and this is his slender, elegant introduction to the field.

Garabedian was a contemporary of John's and is still active at Courant. Nos 18 and 20 are classics in the Intitute's tradition of PDE research. (Like Courant and Hilbert [Nos 23-24] G omits diffusion.)

I may inadvertently short-change this book a bit in my review. It's a solid, detailed treatment of the classic equations and some semilinear variants with an emphasis on calculation. I like it a great deal. As I say in the review, you'll love the exercises--set aside lots of time.

...and C&H-2 treats the Cauchy and Dirchlet Problems. No treatment of diffusion, though, since Fourier had been around for a long time.

This is AMS 108 that I cite in the review. It treats the same topics as the earlier "Linear Monotone Operators" but with different emphasis. They complement each other well. Note that this book and its companion (No 27) deal with both linear and nonlinear functional analysis.

See Sec 10.4, where Z discusses Dirichlet's Principle in electrostatics as a paradigm for mathematical modeling in quantum field theory. Of the three times that Z discusses the intuition underlying the variational approach to PDEs (in Nos 25, 26, amd 28), this is his most developed. A great deal of early 20th-century classical, mathematical physics is contained in only 33 pages. Lovely!

Not only is this where the modern theory began (everything has at least a precursor in Euler!), but a marvelously clear exposition that shows little age. You really can't blame Courant and Hilbert for omitting discussion of diffusion, when F was available. But F has little concern with rigorous analysis of convergence, which occupied an army of mathematicians for the next hundred years,

The most insightful, instructive, and enjoyable science text I've ever used. Period. Opens a whole, new world! Not a hodgepodge--a descriptive title is "Applied Asymptotic and Perturbation Methods for ODEs." Some problems are very, very hard; all are worthwhile. Now if only the authors had done the same for PDEs, as the Preface (sort of) promises....

>> FORTHCOMING << Mark Holmes, author of No 35 and always a delight to read, has a book with Springer due in mid-09, titled "Introduction to the Foundations of Applied Math," that he's pitching as an update of L&S. As of 18 Mar 09 the contents are on his website at Rensselaer (http://eaton.math.rpi.edu/faculty/Holmes/Courses/FOAM/Fall08/Classroom%20Resources/tc.pdf). It looks really good!

Specifically addresses some of the open problems raised in the text (vice exercises) of Van Dyke, so it's useful to read in parallel. Has only a few exercises of its own, though, which is its only defect.