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Can anyone tell me what the reader must already know in order to meaningfully read "Functional Analysis" by F Riesz and Bela Sz Nagy from the start to the very end?

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A little bit of real analysis would help. The book is fantastic, it is clear and the style is polished, however I would not recommend (this is completely personal) to learn about Lebesgue integration from that book. The book is quite self-contained, so if you know a little bit of analysis, enough to understand the first pages you will be able go through it. It might be easier to learn real analysis from other source and jump directly into part 2 of the book which contains all the functional analysis stuff.

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    I'm not aware of a Baby Spivak, if you refer to Baby Rudin, then yes, that should be enough. In the first pages, they start working working with some concepts of measure theory, but they are properly defined without going into detail into the axiomatic part. Perhaps I should warn you that I don't think the book is a popular choice as a textbook neither in a real analysis class nor a functional analysis one. Having said that, the book is wonderful and it shows it evolved from more than 20 years of teaching notes. Hope this helps!2012-10-02