What are some good sources to learn the mathematical background of Quantum Mechanics?
I am talking functional analysis, operator theory etc etc...
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You will have more than enough math for the first two semesters of quantum mechanics if you are taking functional analysis in a mathematics course.
The immense majority of quantum mechanics books will have the requisite math in an appendix. That is true whether or not you want the more sophisticated mathematical treatments (for instance Von Neumman's classic text), or the less rigorous nuts and bolts.
So while you may have to look elsewhere for the theory of distributions (eg the proper way to handle the Dirac delta function) for the most part you don't really need to know it in order to do the physics, or rather it just clutters notation.
This more or less also holds true for Quantum field theory (where eg you need to know some group theory) and again most texts will discuss the necessary material, although of course depending on how far you get it will lead into active research areas where knowing some esoteric maths sometimes does come in handy.
Dover books are good and cheap. For example, here is Mathematics for Quantum Mechanics.
A good place to look for book recommendations for mathematical physicists is this page of John Baez:
I have to agree with the others that the best way to learn the mathematical background of QM is to learn QM, you'll see yourself what kind of mathematical tools you'll have to study further.
Anyway, here are a few tips: If you decide to attend a math course on functional analysis, see that it is about linear operators on Hilbert space leading to the spectral theorem. That's the part of functional analysis you'll need first in QM. "Functional Analysis" is a broad topic, and many math faculties start with more abstract stuff you'll need only later (like Banach algebras or topological vector spaces).
Here is an example of a book that is taylored for the special needs of physicists learning QM:
It is short, it explains everything in detail and covers the essential topics for QM:
Measure and Integration
Distributions, Fourier and Laplace Transforms
Linear Operators, Bounded and Unbounded, Spectral Theory
The most important background is the extension of linear algebra to infinite-dimensional vectorial spaces. So you introduce Banach and Hilbert spaces, $L^p$ and note that only $L^2$ (that's the space of quantum waves functions) is a Hilbert space.
You must study linear operator on $L^2$, and $l^2$: many attentions must be given to adjoint operators, hermitian and antihermitian operators, unitary operators, proiectors...After that, definition of the norm of a vector and a operator and limited and nonlimited operator (a limited operator is a continuous operator), and the Riesz theorem. If you study Lebesgue's measure theory is better.
The definition of tensor product of many hilbert space (only a finite tensor product of hilbert spaces is a hilbert space again) is important too.
Last but not least, Fourier analisys, the notion of a complete and orthonormal basis, scalar products and generalized Fourier series; Green functions.
All these definitions and arguments can be found in this book for example: Reed M , Simon B , Methods Of Modern Mathematical Physics, that is mathematically very rigorous and accurate.