Hilbert Spaces & QM

[mmwave]

I browsed a book by Byron & Fuller "Math. Physics" and read the following:

Algebra, Geometry & Analysis are joined when functions are treated as vectors in a vector space. This makes Hilbert spaces extremely useful in QM.(paraphrased but that's the jist of it)

Comments on this?

If it's true than I am interested in studying Hilbert spaces to better understand QM. When I took QM my teacher was asked "What good is this stuff on Hilbert Spaces?". His reply was that every space & every function used in QM is a Hilbert Space so don't worry about it - it's of no consequence at all.

[selfAdjoint]

The states of a quantum are represented numerically in a Hilbert space, which is a mathematical object. The simplest Hilbert space, for the simplest quantum systems, is the complex numbers. Only a little more compicated are sets of vectors with complex components. In field theory the vectors have infinitely many components, like the coefficients of a Fourier series.

Things the experimenters do are represented by mathematical operators that act on the vectors and turn them into other vectors. Sometimes an operator only multiplies a vector by a number. In that case the vector is called an eigenvector of the operator and the number is called an eigenvalue. There are some operators, called Hermitian operators, for which the eigenvalues are always real (as opposed to complex) numbers. All the genuine operators in quantum mechanics are assumed to be Hermitian. The eigenvalues represent things that can be measured in experiments. They are called observables.