The Analysis of Linear Partial Differential Operators I

Vol. 1 of Lars Hörmander's influential 4-volume treatise is a detailed exposition of the theory of distributions.

From the reviews:

"In order to illustrate the richness of the book: in my review of the 1983 edition [...] I gave a list of 20 subjects which were new compared to Hörmander's book of 1963. Most of these subjects concern important, basic and highly nontrivial theorems in analysis. Hörmander's treatment of these is extremely clear and efficient and often highly original. [...] Most of the exercises are witty, with an interesting point. The phrasing of both the exercises and the answers and hints is very careful [...] In all, the book can be highly recommended, both as a textbook for advanced students, and as background and reference for introductory courses on distributions and Fourier analysis."

J.J. Duistermaat in Mededelingen van het Wiskundig Genootschap