Functional analysis book, Quantum mechanics

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Which book do you recommend to get a rigorous background in functional analysis ? I want to after that study quantum mechanics. Should I look into grandpa Rudin?

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There are many good introductory functional analysis books and several questions covering this topic. The choice is mainly a matter of personal taste, just be sure that the spectral theorem for unbounded self-adjoint operators is covered in detail.

If your main goal is to study quantum mechanics however, there are several books that take some shortcuts to deal mainly with the topics (they deem) necessary for quantum mechanics:

  • Prugovecki. Quantum Mechanics in Hilbert Space
  • Teschl. Mathematical Methods in Quantum Mechanics
  • Weidmann. Linear Operators in Hilbert Space (there is also a second part, but apparently only in German)

All these books can be read without prior knowledge of functional analysis (although they may cite some theorems from Banach space theory once or twice) and lead up to discussions of some mathematical aspects of quantum mechanics.

The price you pay for the short path is that you won't learn much (or anything) about Banach spaces and linear operators on them, which is a core part of classical functional analysis. So if your main objective is to acquire a solid knowledge of functional analysis, this might not be the way to start.