There are a lot of coherent definitions of Hilbert spaces. Lets take the wikipedia one and let me ask you some cuestions: A Hilbert space: 1) Should be linear 2) Should have an inner product (the bra - ket rule to get an amplitude) 3) Should be complete (every cauchy sequence should be convergent) Why states should have these properties? Thanks!!! Pd: I would like to receive answers that are not dependent of the results of QM. For example, we can make an argument that starts by accepting the Heisenberg principle and from this, perhaps, we can derive that the state should be represented by an element of a Hilbert space. But I want to go all the way around. I would like to prove or explain by really basic hypothesis why 1), 2) and 3) should hapend and then, for example, prove Heisenberg principle (adding first, all the other stuff such as the Born rule, evolution equation and such).