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I am in search for Functional Analysis Books.

I need a book which have lots of elementary problems.

I mean it should force us to think more which will give a deeper insight for functional analysis.

To be precise, I need lots of problems like Is $A$ open in the space $\ell^1$ with respect to norm $\|.\|_2$. Find a counterexample to this statement and etcetera.

I mean it shouldn't ask us trivial questions. It should ask us questions which will force us to think more and more from functional analysis

Theo Bendit
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GraduateStudent
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    Sheldon Axler, author of the popular book Linear Algebra Done Right, has a new book called Measure, Integrations, and Real Analysis that contains an introduction to functional analysis. The book is free on Axler's web page. It's worth a look. Many exercises. – littleO Sep 03 '19 at 07:53

6 Answers6

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The most elementary book for functional analysis I've seen is Introductory functional analysis of E.Kreyszig, really really understandable and its problems are 'basics'.

user1trill
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John B. Conway: A Course in Functional Analysis (Springer).

Harro G. Heuser: Functional Analysis (Wiley & Sons).

A.E. Taylor and D.C. Lay: Introduction to Functional Analysis (Wiley & Sons).

Fred
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I think that the book that you need is "Lectures and Exercises on Functional Analysis" by Aleksandr Khelemsky. This book has the following merits:

1)It teaches the modern approach to the Functional Analysis based on Linear Algebra, Axiomatic Set Theory and Category Theory.

2)It gives information not only about classical Functional Analysis, but also about some more advanced topics such as Quantum Functional Analysis and Banach Homological Algebra.

3)The exercises there are basically small lemmas, that you need to prove by yourself before proceeding further. They are interesting and fun to solve, and have direct connection to the context in which they appear.

4)The theory is built gradually from the very basics to the more advanced topics and everything is proved strictly. This makes the book suitable even for beginners.

5)The book is very fun to read and the writing style of the author is amazing. When you start reading it, you will not be able to stop.

Chain Markov
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As I can see there are very good answers by various members . After referring to them you can successfully refer to :

K. Yosida : Functional Analysis

It's good book and requires prerequisites given by other members. It's relatively advanced w.r.t the given answers.

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You might be looking for something like

Costara and Popa: Exercises in Functional Analysis

I cannot comment too much, though, because I did not read it.

Tom
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If you're new, try Simmons's book. Other than that both books of Walter Rudin and John B. Conway consists of many non-trivial questions. P.R. Halmos's "A Hilbert Space Problem Book" by springer is difficult to me atleast.

Achak0790
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