Complementary text for mathematical Quantum Mechanics lectures I'm looking for a text to complement Frederic Schuller's lectures on QM.  His approach is very mathematical -- in fact it looks like the first 12 of 21 lectures are just about the mathematical foundations of QM.  He introduces Hilbert and Banach spaces from scratch (from basic linear algebra and analysis really), derives the functional analysis version of the spectral theorem, and gives what I assume are more rigorous definitions.  For instance in all of the undergrad books I've looked at, quantum states -- if they're given any definition at all -- are said to be elements of the Hilbert space.  But Schuller claims that that is not correct.  States are in fact positive trace-class linear maps on the Hilbert space.  Does anyone know a good undergrad level QM book that I can follow along with so I have some exercises and extra material to work through as I go through the lectures?  Thanks.
 A: I'll just make my comments into an answer.


*

*Reed and Simon Volume 1: Functional Analysis (Methods of Modern Mathematical Physics) here on amazon has a table of contents. 

*Reed and Simon Volume 2: Fourier Analysis, Self-Adjointness (Methods of Modern Mathematical Physics) here on amazon, again you can see toc.

*Cohen-Tannoudji - Quantum Mechanics (2 vol. set). Reasonably rigorous and may fit Schuller to some extent - lots of end of chapter appendices. amazon link

*Messiah Quantum Mechanics (2 Volumes in 1) - two volumes has a lot in it, might not be as rigorous as you want it. amazon link

*Quantum Mechanics in Hilbert Space: Second Edition, suggested by user254665 - The preface to the first edition starts "This book was developed from a fourth-year undergraduate course given at the University of Toronto to advanced undergraduate and first-year graduate students in physics and mathematics. It is intended to provide the inquisitive student with a critical presentation of the basic mathematics of nonrelativistic quantum mechanics at a level which meets the present standards of mathematical rigor." Seems to fit the course reasonably well judging by toc - amazon, toc, preface etc.
R&S Volume one introduces Hilbert spaces, Banach spaces, spectral theorem etc. and leads from bounded to unbounded operators and the fourier transform.
R&S Volume 2 is very physics orientated, with topics on fourier transforms, hamiltonians in non-rel QM, and talks about self adjoint operators, and a bit about time dependent Hamiltonians.

As a note on quantum states, there's various definitions. They can be 


*

*vectors in a Hilbert space $\mathcal{H}$, but specifically one that satisfies a Schrodinger equation you're interested in (Assuming you're in Schrodinger picture in non-rel QM). State space would be a subspace of a Hilbert space.

*elements of the projective hilbert space $\Bbb P\mathcal{H}$

*traceclass positive operators of trace $1$, normally called $\textit{density matrices}$.

*A rank one projection operator.


It depends on what you want to do with them.
The first I think is the most common, as when teaching quantum mechanics, the wave functions usually belong to an $L^2$ space, and are found by solving the Schrodinger PDE.
The second last one is useful for statistical mixtures and open quantum systems, you can have pure and mixed states. The pure states can be identified with the last item on the list.

As a sidenote this was asked by a different user on physics stack, same question on schullers course, and it was closed, even though it's physics, and a pretty reasonable request. It might be useful to check there for the one answer that was able to be posted before it was closed.
https://physics.stackexchange.com/questions/259583/good-texts-on-quantum-mechanics-to-accompany-this-online-course#comment579079_259583 
