Hilbert space and eigenvalues

I suppose I don't understand your question. A Hilbert space is an inner product space in which all Cauchy sequences converge (which is really just a pedantic/rigorous way of saying that it is complete). In physics we often say that quantum states "live" in a hilbert space (although to be accurate it's actually a 'rigged' Hilbert space since plane waves and dirac delta functions are not in a Hilbert space). A valid quantum operator must be an eigenvector of a quantum state with a real eigenvalue. If none of this means anything to you then I'd say to get yourself a good linear algebra book. (If you have no idea what linear algebra is I'd say wherever you read about Hilbert spaces and eigenvalues is probably way too advanced and I'd suggest starting with a more introductory book)

 

It's never too soon to study linear algebra. I have a vague recollection that Galileo said that if he had his life to live over again, he would first have become a mathematician.

Strang's book is one of the standards. I have not read it, but I'm told that it emphasizes computation and the ability to solve problems numerically. Personally, I'm fond of the book by Bamberg and Sternberg, even though it's a little tough for self-study.