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I took a quantum mechanics class for the first time this past semester, and didn't feel like I fully grasped many aspects of the subject, so decided to try to go through some stuff on my own, and just had a gew questions.

One of them is more a math question: In the book I'm reading (Quantum Mechanics by Zettili) they mention that the space of all complex functions is infinite dimensional, but don't state why, and I'm not quite sure how to show something like this. I know that I need to show that there are an infinte # of linearly independant basis vectors, but I'm not quite sure how to get at this.

My second question is about Hilbert space. This is something else that I don't fully grasp. It seems like something very important to quantum mechanics, but I'm not quite sure why the hilbert space is a good stage for quantum mechanics. The only things I could think of were that maybe it has something to do with how the inner product of some vector in the hilbert space with itself is real (I know that this is the reason why hermitian matrices are important, because the eigenvalues are real, and I thought maybe it was similar), or perhaps something about the whole limit of cauchy sequences coverging.

I guess this is a the same question sort of but what are spaces that are not hilbert spaces and what about them is not useful in the same way that hilbert space is?

So am I onto anything or way off? Any help would be greatly appreciated