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I'm in search of a good book that I can read --- and recommend to my proteges to read --- along with each one of the following books.

  1. Topology by James R. Munkres, 2nd edition

  2. Introductory Functional Analysis With Applications by Erwine Kreyszig

  3. Principles of Mathematical Analysis by Walter Rudin, 3rd edition

For each one of the above three books, I would like to have a book (or a couple of books) that discusses the same material as in this book and which fills the gaps in the presentations of the topics covered herein by illustrating further some or most of the topics discussed rather cursorily in this book, providing more examples on the topics of this book, and even providing the solutions to the exercise problems in this book as text examples. In short, in each case, I would like to have a book (or a couple of books) that reinforces one's understanding of the contents of this book.

And, last but not least, I would also like an introductory-level, and preferably recently written / updated, functional analysis textbook that is accessible to a reader with the real analysis background from Introduction To Real Analysis by Bartle & Sherbert, or from Baby Rudin's first seven chapters.

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  • $\begingroup$ Each of these books has several mathematics stack exchange posts devoted to it. I recommend looking for those (use google, along with "Stack Exchange" in quotes) to see what other books people mention when discussing the books you mentioned, as well as what you can find in amazon.com reviews and the like. Keep in mind that most people will only have more than a passing knowledge about 2 or 3 books related to each of these, so suggestions taken individually should be viewed with caution. (continued) $\endgroup$ – Dave L Renfro Feb 19 '18 at 23:21
  • $\begingroup$ What you want is to look at are averages and trends in the comments made about books. Personally, I would just browse a university library's holdings, but that's because I usually know what I want when I can see it and judge it for myself. (moments later) I just read you profile, and perhaps browsing a university library's holdings is not something you can do. Nonetheless, it is something others seeing your question can do, so I'll leave my comments in place for now. $\endgroup$ – Dave L Renfro Feb 19 '18 at 23:22
  • $\begingroup$ I got the 4 book series in analysis in the Princeton series( for example amazon.com/Complex-Analysis-Princeton-Lectures-No/dp/0691113858). I'm hoping they do some of what you want... I may find out next year. It depends on the focus of my elite students here... This is a great question I hope you get a complete answer. $\endgroup$ – James S. Cook Feb 21 '18 at 3:03

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