Question

How to construct Hilbert spaces which are needed for quantum mechanics?

Answer 1

If you want to learn Hilbert spaces from a mathematical point of view, unfortunately it sounds like there is a long road to travel. If you haven’t taken an analysis course, you absolutely need that knowledge. (Not necessarily saying you need to take the course, but you should do the equivalent, whether it’s by independent study or something else.)

Then, you need to learn basic functional analysis. Only then can you meaningfully start to study Hilbert spaces.

But there’s good news. If you really just want to learn enough about Hilbert spaces to explore QM, then you don’t need as solid a mathematical background. In fact, I boldly claim you don’t even really need to know what a Hilbert space is… just that there’s this perfectly natural notion called an inner product, and it’s used for… ummm… doing operator stuff on wavefunctions or something like that.

Perhaps a good analogy is this: you learn somewhere that hockey is played on an ice rink. You want to play hockey. That doesn’t mean you have to become an expert on how to build an ice rink… you just sort of have to know how to use it.

Hilbert spaces are the ice rinks of quantum mechanical hockey.