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I recently started functional analysis in more depth. I followed a standard course on functional analysis at the master's degree but clearly it was very basic and many results are deeper than the one it covered.

I'm looking for a book which starts from topological vector spaces and goes through to metric spaces, normed spaces, Banach spaces and Hilbert spaces. IN this sense [Rudin, Functional analysis] is surely the right book. Are there some simpler alternatives to Rudin's book with this structure so that I can read 2 or more books?

(Also [Yosida, Functional analysis] has this kind of structure but it is too advanced and very difficult to read for me).

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  • $\begingroup$ A book that starts out general (topological groups, then topological vector spaces) and only later gets to more specialized settings (normed/Banach spaces, then inner-product/Hilbert spaces) is Lectures in Functional Analysis and Operator Theory by Sterling Khazag Berberian (1974). Although I've had my copy since the late 1970s, I've never dealt with it much (only occasionally over the years looking up something in it), so I don't know how suitable it would be for your purposes. $\endgroup$ Dec 10, 2022 at 13:55
  • $\begingroup$ @DisintegratingByParts what is the one which has most intuitive proofs and that explains them as much as possible? $\endgroup$
    – carlos85
    Dec 13, 2022 at 9:56
  • $\begingroup$ Thanks to all for the answer by the way $\endgroup$
    – carlos85
    Dec 13, 2022 at 9:56

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Here is a classic that starts with topological vector spaces and works down. It's a Dover publication.

I think this is what you're looking for.

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