A compilation of reference books

Mathematics

Functional Analysis

  • Functional Analysis, F. Riesz and B. Nagy. 1955 (1990). According to one source the book includes a first part which is mostly included in modern real analysis textbooks, and a second part treating integral equations, Hilbert and Banach spaces, and spectral theorem. It is described as clear and polished, self-contained, though without exercise.
  • Introductory Functional Analysis with Applications, E. Kreyszig. 1991. Seems to be the go-to book for non-mathematical-major engineers/physicists. Less dull or rigor compared to Rudin, and oriented towards applications.
  • Functional Analysis, P. Lax. 2002. This is a standard textbook by the Abel Prize laureate. Less challenging than Rudin’s book.
  • Funktionalanalysis, D. Werner. 2005. A German textbook by D. Werner at Freie Universität Berlin. Given that it is only offered in German, it is unusual that it has been recommended in at least three independent sources online.
  • Functional Analysis, Sobolev Spaces and Partial Differential Equations (Analyse fonctionnelle, theorie et applications). H. Brezis. 2010. Originally in French by Brezis, later translated into English. It is described as elementary, self-contained, and has all the exercises as well as partial solutions. Might be very nice for self-studying.
  • Functional Analysis: Introduction to Further Topics in Analysis, E. M. Stein and R. Shakarchi. 2002 (2011). Recommended by Prof. Marc Burger at ETH. This belongs to a four-book series. The previous one on real analysis might be a good starting point if one wants to refresh his/her memory on (or like me, start to learn about) measure theory, etc.

From this list, it seems that continental Europe has somehow hosted the most of the masters in functional anlaysis. Riesz and Nagy are Hungarians; Peter Lax born in Hungary, although educated in US since high school years. Dirk Werner is German; so is Kreyszig in his years of education. Haim Brezis is French. Perhaps this belongs to part of the European tradition of analysis.

There is also some recommendations for Kantorovich’s Functional Analysis in Normed Spaces and Kolmogorov’s “Elements of the Theory of Functions and Functional Analysis”, which are probably also worth are read. Too many materials are worth reading, but life is too short. 所谓“哀吾生之须臾，羡长江之无穷”也。