9

There is Methods of Modern Mathematical Physics by Reed and Simon, which is a 4-volume book which teaches functional analysis, with a focus on operators in Hilbert spaces. Its main aim is to provide a sound mathematical background for the methods used in quantum mechanics, but it serves well as a textbook on functional analyis (for example, it covers some ...


8

At the introductory level, Erwin Kreysiz's Introductory Functional Analysis with Applications is excellent. See the reviews at amazon.com. It's probably a bit elementary for you at this point (still, it could be very suitable for others reading this thread for recommendations), but I recommend at least looking through a copy at the library. I've actually had ...


6

I am, perhaps, late to the party, but this question has no answer, so here are a couple of my own suggestions: Stein, Elias M.; Shakarchi, Rami, Functional analysis. Introduction to further topics in analysis, Princeton Lectures in Analysis 4. Princeton, NJ: Princeton University Press (ISBN 978-0-691-11387-6/hbk; 978-1-400-84055-7/ebook). xv, 423 p. (...


6

Peter Lax: Functional Analysis. It is sometimes difficult to use because of its cryptic style, but it is a great source of applications and a great source of historical references on applications which motivated functional analytic concepts.


6

Look for "Radon integrals". I am not sure about any advantages, but the book by Gert Pedersen: Analysis now contains a chapter (Chapter 6) on integration theory. This is I think in the style you want: This chapter has two functions: Throughout the book it has served as an Appendix, to which the reader was referred for definitions, arguments, and ...


5

Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations. The book is a great introduction into functional analysis but it is built up to show how FA results can be used to solve partial differential equations. So if you count PDEs as applications, then this is a great book.


4

If you want to study real and complex analysis, a classic text is W. Rudin's "Real and complex analysis". In fact pretty much anything by Rudin is excellent. A more accessible title of Rudin is "Principles of Mathematical Analysis". Another classic complex analysis book is by Alfors, although the title by Stewart and Tall, both called "Complex analysis", ...


4

If you know German, then the book by Harro Heuser is the one you are looking for. It contains many many applications and mixes the theory with relevant applied material. An English translation of an earlier edition is available from Wiley.


4

Students should learn many versions of the development of "integral". Simultaneously, we should ask "why bother?" A Lebesgue "formal" theory of integration, or the possibility of such a development (even on spaces without a topology) is interesting. However, mostly we don't care about such situations; the Riesz representation theorem explains most common ...


2

A would agree with Rusan Kax but like to add that another good introductory analysis book is by Tom Apsotol. Also, for beginners in analysis, I would say either start with Rudin's baby analysis or Apsotol. As for Complex analysis, you may want to learn some abstract algebra and/or number theory since many texts incorporate the use of Algebra. Books in ...


1

Worth a mention is Pons' Real Analysis for the Undergraduate: With an Invitation to Functional Analysis. While it is primarily an introductory real analysis text, the final section of each chapter is a functional analysis topic. For students who have already had a light introduction to real analysis, you could still make use of these final sections and ...


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