I am a student of mathematics attending a course in Quantum Mechanics. This course is held by a physicist, and it is really confusing for me to follow his reasonments. With this, I do not mean to be arrogant or disrespectful, it is only a matter of backgrounds. I was wondering if anyone knows a textbook which is mathematically rigorous, in the sense that the functional analytical structure of the subject (distributions, measures, Hilbert spaces, duality) is properly handled. Instead of doing inconceivable (for me) things such as considering Dirac's delta a function or distinguishing between a state in $L^2(V)$ and its Fourier coefficients in $l^2$, when we have a nice isomorphism between these two.
I have background in Functional Analysis, Measure Theory and Topology, but a reference which explains also some mathematics would be good, you never know. The course covers all the basis of quantum mechanics, hence: Schroedinger and Dirac's picture with examples (hclassical problems, harmonic oscillator, hydrogen atom), spins and reaches the helium atom. Mostly it follows Schwabl Quantum Mechanics https://www.springer.com/it/book/9783540719328, which I consulted but does not provide what I search for.
I really want to make it clear: I really appreciate physicists and their stunning mind elasticity: after all, most of distribution theory was developed to underpin what Dirac had grasped with intuition. But I am just a student trying to do my best at college, and I need to cope with my background!
Thanks in advance.