Complementary book for Conway's book "A Course in Functional Analysis"

Question

I am first year graduate student and I have taken Functional Analysis course this semester. The main book of teaching is Conway's book "A Course in Functional Analysis" but the lecturer is notorious for giving exams from questions that are not exercises of the book he's teaching. So I need an accompanying book for functional analysis with lots of cool problems that covers same materials of Conway's book and is on graduate level. Any suggestions would be much appreciated.

Answer 1

As people have recommended in the comments, Rudin is a good book for exercises and as a complementary book, but I would not recommend studying from it. Also, Kreyszig's book is a standard functional analysis textbook: don't underestimate it just because it is considered undergraduate. Graduate students can learn much from this book too.

Two undergraduate level textbooks with cool aspects of functional analysis and nice, pedagogical exercises are the following:

• Barbara MacCluer, Elementary Functional Analysis https://www.springer.com/gp/book/9780387855288

• Karen Saxe, Beginning Functional Analysis https://www.springer.com/gp/book/9780387952246

Keep in mind that they are aimed at undergraduate students, so they might not be really close to what you are looking for. But, for example, the standard topics of "The big three theorems" or the Hahn-Banach theorem are nicely addressed. Moreover, there are many important exercises included that someone might have missed in their first functional analysis course.

Of course, Conway's book is nice to study from but I cannot not recommend G.K. Pedersen's book "Analysis Now": https://www.springer.com/gp/book/9780387967882

A book with lots of information about standard Banach spaces and their "geometry" and quite a few exercises is N. L Carothers's "A Short Course on Banach Space Theory": https://www.cambridge.org/core/books/short-course-on-banach-space-theory/88B71D535E744B24CA17CA82E670BFF4 but I would only recommend having this "on the side".

And, a book with many, difficult (this is only my opinion of course) exercises on $L^p$ spaces (and measure-theory), Banach spaces, Hilbert spaces and operators between those is Alberto Torchinsky's "Problems in Real and Functional Analysis": https://bookstore.ams.org/gsm-166 The book comes with the solutions to half the exercises. As I said, I think that these are difficult problems so do not fall in the trap where you start solving one problem, get stuck and then fixate on solving it!

And, as mentioned in the page between the first part of the book and the solutions to the exercises:

Don't panic!

Answer 2

Below are some books that I use sometimes:

• "Functional Analysis" written by Theo Bühler and Dietmar A. Salamon: https://people.math.ethz.ch/~salamon/PREPRINTS/funcana-ams.pdf
• "Topics in Linear and Nonlinear Functional Analysis" written by Gerald Teschl: https://www.mat.univie.ac.at/~gerald/ftp/book-fa/fa.pdf
• "Linear Functional Analysis" written by Hans Wilhelm Alt: https://link.springer.com/book/10.1007/978-1-4471-7280-2
• "Functional Analysis - An Introductory Course" written by Sergei Ovchinnikov: https://link.springer.com/book/10.1007/978-3-319-91512-8
• "An Introductory Course in Functional Analysis" written by Adam Bowers and Nigel J. Kalton: https://link.springer.com/book/10.1007/978-1-4939-1945-1

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The book I most recommend is the following:

• "Functional Analysis for the Applied Sciences" written by Gheorghe Moroşanu: https://link.springer.com/book/10.1007/978-3-030-27153-4

Although there is "for the Applied Sciences" in the title of the previous book, the book is rigorous and very good for beginners. The theorems have proofs and almost all exercises are solved at the end of the book.