>>7793092 You should know some basics before you start in particular real analysis, linear algebra (with respect to vector spaces) and some elemtary concepts from (point set) topology otherwise you might have to look up a lot of things while trying to get into functional analysis itself. If you ´re looking for a book to start with, I´d recommend Yoshida as a classic or Werner (but I don´t know if there is an english version of it). I also really like the writing style of Brezis but I only ever read his book and functional analysis and PDEs not the one about functional analysis.
>>7793355 >Is abstract algebra needed? For an introduction no. But once you get to more advanced topics like representation theory on Hilbert spaces and operator algebras, you'll need some algebra background.
The progression I would suggest is the following: real analysis, linear algebra, complex analysis and measure theory. Then find a book, I used foundations of modern analysis by avner friedman. It's pretty compact tho, written for 3rd year mathematics students. Someone suggested Introductory Functional Analysis by Erwin Kreyszig, I haven't used it but I heard good things...
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