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I am a masters 1st year student and my functional analysis course ended 1 week ago. The instructor followed Irwin Krieszig's Book " Introductory Functional Analysis with Applications" for teaching and although he didn't gave any assignments or exercise problems, I solved problems from exercises given in Krieszig's Book. I was able to understand the concepts given in the book clearly and then do exercises of Krieszig without much difficulty. But I noticed that the problems of the book are much easier for a masters student but as I was busy in some extra - curricular activities also , so I didn't tried other problems or asked for reference books or materials here.

But Now I think that I must solve good number of Challenging problems in Functional Analysis as I will apply for phd in pure mathematics soon and I got my December break time to it.

So, Can you please tell any book in functional analysis course whose exercises are challenging as compared to Krieszig's / any assignments which any professor of a university gave which are available on web?

Also, I am adding my course syllabus in the following images so that you come to know exactly which topics should be available( I absolutely no problem If more topics are covered). All these topics were covered which are given in the images and none except them were covered.

Edit: I am good in measure theory , so if any functional analysis book you are recommending assumes knowledge of measure theory then I am fine with it.

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Looking for your help.

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    $\begingroup$ A nice book on Real Analysis by Sheldon Axler] (It has chapters on functional analysis also) is available free online. $\endgroup$ Commented Dec 2, 2020 at 13:42
  • $\begingroup$ Take a look to Exercises in analysis (Part I) by Leszek Gasiński, & Nikolaos S Papageorgiou and also Exercises in Functional Analysis by Constantin Costara & Dumitru Popa. They have solutions and good summary of contents. $\endgroup$ Commented Dec 2, 2020 at 13:44
  • $\begingroup$ A "challenging" book is Dunford and Schwartz. But I wouldn't read it. $\endgroup$ Commented Dec 2, 2020 at 20:22
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    $\begingroup$ @Ben: I appreciate Kreyszig's book, it's a nice and highly readable classic. But note that Kreyszig explicitely excludes measure theory. If you are not comfortable with measure theory you might have difficulties, since many functional analysis books assume a basic knowledge of measure theory as prerequisite. You might want to address this aspect in your question to get appropriate book recommendations. $\endgroup$ Commented Dec 3, 2020 at 19:55
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    $\begingroup$ @MarkusScheuer Ok, I have edited my question to mention that I have already done a course on measure theory. $\endgroup$ Commented Dec 4, 2020 at 11:47

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You may follow the following references:

$1.~~$ "Theorems and Problems in Functional Analysis" by A. A. Kirillov, A. D. Gvishiani
$2.~~$ "Problems in Real and Functional Analysis" by Alberto Torchinsky
$3.~~$ "Theorems and Problems in Functional Analysis" by J. D. Knowles
$4.~~$ "A Glimpse at Hilbert Space Operators" by Paul R. Halmos

These books contains many standard problems. Most of the problems are hard but very good once it is done. I think you may get benefit from here so just try it.

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I'll recommend Exercises in Analysis, Part I by Gasiński, and Papageorgiou. It's a brilliant source of thoughtful and challenging exercises in mathematical analysis; starting from basics of metrics spaces, then topology, measure theory and functional analysis. Also, it comes with full solutions.

For more advanced topics there's also a second part too by the same authors.

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