For anyone who wants to study quantum mechanics and has a math background, I strongly recommend Shankar's quantum textbook. It has the most rigorous, clear, and correct derivations I have seen thus far, starting with vector spaces and moving to physical systems only after a hundred or so pages.
The book is just down right atrocious. The fact that it became almost standard in American universities speaks more of the power of networking than anything else. Shankar's book is anything but rigorous, quite on the contrary.
There are so many good QM books:
For an absolute beginner I'd recommend David Griffiths
Dirac's is very readable
Sakurai's is very good though a bit advanced
Whatever you do, it's difficult to go wrong by not using Shankar's
For someone a little more knowledgeable it might get a bit boring...
Haven't read Griffiths yet so I can't comment on it.
Griffiths doesn't even get into Hilbert spaces until chapter three, at which point he delivers a broken and incomprehensible explanation. He attempts to teach Hilbert operators by analogy to the multiplication or application by adjacency people use in normal arithmetic. This is a terrible approach, and confuses the hell out of students who haven't already taken abstract algebra.
In what way is Shankar less rigorous than Griffiths?
If you've got a math background I'd recommended you skip Shankar even more vehemently than I usually do.