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I recently started studying functional analysis. I have many ebooks loaded on my laptop, but can't figure out which one to start with. I've asked my instructor, and he says there aren't any specific books to start with. Here's a brief list of the topics on our syllabus:

  1. Normed linear spaces
  2. Banach Spaces
  3. Hilbert spaces
  4. Compact Operators
  5. Knowledge of $C[0,1],L^p[0,1]$
  6. Continuous linear operators
  7. Hahn-Banach theorem
  8. Open mapping and Closed Graph Theorem
  9. Uniform Boundedness Principle

I'm seeing these topics first in my lifetime. One thing I want to ask is whether functional analysis is very difficult. Please suggest some books to start this topic which covers the above and contains some exercises. Thanks in advance.

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  • $\begingroup$ I have personally been reading Metrics, Norms and Integrals An Introduction to Contemporary Analysis by JJ Koliha for fun in my spare time. It seems to cover most of what you listed, but if anything I would argue it is too easy to read. I switch over to Erwin Kreyszig's Introductory Functional Analysis with applications when I think I've learned something from Koliha and need a lesson in humility. $\endgroup$ – JessicaK Jan 28 '15 at 5:14
  • $\begingroup$ For an introductory course, my recommendation is to go for a book with a lot of exercises. The first book I ever used was a book by V. A. Trenoguin, B. M. Pisarievski, T. S. Sóboleva, Exercises in functional analysis. Of course, self study with more advanced/reference books is important later: Dunford-Schwartz, Yosida's, Halmos', Halmos', Halmos'... Oh, Conway's is nice too. Komogorov-Fomin (!?) $\endgroup$ – Pp.. Jan 28 '15 at 5:29
  • $\begingroup$ I like Kolmogorov & Fomin. $\endgroup$ – copper.hat Jan 28 '15 at 5:49
  • $\begingroup$ Kreyzig is a popular and friendly introduction to Functional Analysis that covers the topics you listed. $\endgroup$ – littleO Jan 30 '15 at 9:33
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You can start out with the book Introductory Functional Analysis by B. Daya Reddy.

I started reading it and was not able to stop because everything was so clear and rigorously defined.

This book is unlike any other book you find off of the shelf, which are filled with indecipherable notations that are not transferable beyond what that particular book teaches you. Instead, the book by Reddy is filled with practical yet simple exercises that you can deal with only reading what is contained in the text and nothing else. What is so amazing is that the material contained in this book is so transferable, after reading it you will be able to tackle any research paper at least at the engineering level.

It is truly the best book to start out functional analysis

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My functional analysis course's Textbook : Functional Analysis , K.Yosida.

This book contains a wide range of topics in FA, and many proofs are elegant.

The only shortage is that there is no exercises in the book.

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    $\begingroup$ Good book! I love it. But it is not a book to begin with. No exercises. Very good for reference, though. $\endgroup$ – Pp.. Jan 28 '15 at 5:22
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Real and Functional Analysis by Serge Lang.

Functional Analysis by Walter Rudin.

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The very best introductory text on functional analysis that exists is the classic Functional Analysis by Bachman and Narici. Using a background of only basic linear algebra and advanced calculus/elementary real analysis/honors calculus, the authors develop all the standard topics on function spaces clearly and simply, with many examples and exercises. Best of all,it's available from Dover,so it's very inexpensive! To me, this is the very best self-study text that exists on the subject and I'd heartily recommend it to any beginner.

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  • $\begingroup$ Another point off for no apparent reason. $\endgroup$ – Mathemagician1234 Jul 19 '18 at 3:24

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