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I've for some time been learning elementary functional analysis from the book Introductory Functional Analysis With Applications by Erwine Kreyszig.

Although Kreyszig's presentation is mostly accessible for a person with not-a-very-strong background in analysis, there often are occasions where, as I feel, he sacrifices rigor and detail in the interest of simplicity of the presentation. And, sometimes he is too verbose; sometimes he is perhaps not explicit enough. Besides, Kreyszig's text, as far as I know, is not being followed at those "elite" places like the MIT who offer such anciliary support as the MIT OCW.

So I would like to have another text that covers the same (or roughly the same) material as does Kreyszig (and preferably a text that covers this material in a more rigorous and explicit manner).

Can the esteemed Math SE community suggest any such text?

Is there any place on the Internet where I can access notes of courses taught using Kreyszig's text, especially the first four chapters?

For those who haven't got a copy of Kreyszig, here is the link to a page where it can be accessed (and perhaps can also be downloaded) for free.

http://www-personal.acfr.usyd.edu.au/spns/cdm/resources/Kreyszig%20-%20Introductory%20Functional%20Analysis%20with%20Applications.pdf

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  • $\begingroup$ Conway's book on the subject includes a lot of information, but it leaves a lot of proofs and proof details to the reader, and if you don't have some experience with topology and measure theory you may get lost in the details. The dover text by Bachman and Narici would be somewhere between Kreyszig's and Conway's, but it is somewhat aged in its presentation (only mentions weak and weak$^*$ convergence, not the underlying topologies, and compact operators are referred to by their older name, completely continuous operators). $\endgroup$ – Aweygan May 17 '17 at 14:54
  • $\begingroup$ Check Hunter and Nachtergale: math.ucdavis.edu/~hunter/book/pdfbook.html $\endgroup$ – Moishe Kohan Sep 4 '17 at 21:33
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The following are standard (often classical) references in functional analysis that cover the area at a more advanced level than Kreyszig:

  1. Rudin
  2. Yosida
  3. Dunford & Schwartz
  4. Reed & Simon
  5. Lax
  6. Conway
  7. Schechter
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