Asking for reference for challenging problems for Functional Analysis 1st course

Question

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.

Looking for your help.

Answer 1

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.

Answer 2

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.