What is a good introduction to Hilbert space theory?

Question

What is a good gentle introduction to the theory of Hilbert spaces for physicists? I mean something that also has some examples interespersed between the theorems and proofs, and shows a bit how to actually calculate things :-).

I personally prefer materials that are a bit smaller than a whole heavy textbook, like lecture notes for example, but of course I appreciate anything that is useful.

Comments

See if you like this one - "Spectral theory of operators in Hilbert space" by K. O. Friedrichs. I have read some portions from it and liked it.

Answer 1

I like Prof. Tan's (Brown U.) gentle introduction to Hilbert spaces. I always refer physics students to it. Here is the description of the notes in his words: "We will not present an exhaustive “mathematical” discussion of this subject. Rather, by using examples and analogies, hopefully you will feel more “at ease” with “Hilbert space” at the end of this short discussion." 

Answer 2

I like this introduction and also this one which is quite sort as you want it. I guess you can find useful material in the classic textbook of Sakurai and maybe it is also useful to check out the book Hilbert Space Operators in Quantum Mechanics , although I have not looked at this in great detail (only some paragraphs). Now, I guess you can go much more mathematical after but I do not know  any better references (