Hello! I have noticed that most advanced textbooks on QM start the development of the subject with a long review of linear algebra. In particular, they talk about pre-Banach, Banach, pre-Hilbert, Hilbert spaces and so on. Why is it necessary to invoke such abstract spaces in order to describe the physical reality? I mean, for example, why do you need that every Cauchy sequence converges within the space to have something physically meaningful?