I'm looking for good references to learn about the mathematical foundations of quantum mechanics. By mathematical foundations, I do not mean rigorous quantum mechanics in general but the axioms behind it from a mathematical point of view: the definition of a state, mean value of operators, representation of states using the spectral theorem and so on. In special, I'd like to see some discussions on the relation between states (in Dirac notation) and wave functions. Most references I know (from the mathematical point of view) discuss the foundations only using wave functions and $L^{2}$ spaces and no connection to the actual general picture is provided.

**EDIT:** Maybe I should express myself a little better to avoid confusion. I know some books on quantum mechanics from a mathematical point of view e.g. Gustafson and Sigal's book. However, these references usually avoid the axioms or discuss them very briefly and the main object of study become wave functions living on $L^{2}$ spaces, where the Schrödinger equantion is solved for a bunch of different potentials and so on. I'd like to have some nice presentation on the axioms itself and how quantum mechanics arises from them in a more systematic way.

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