What is the dimensionality, N, of the Hilbert space (i.e., how many basis vectors does it need)? To be honest I am entirely lost on this question. I've heard of Hilbert space being both finite and infinite so I'm not sure as to a solid answer for this question. Does the Hilbert space need 4 basis vectors to generate the 4x4 matrix of spin orientation (up up, up down, down up, down down). Could someone please clarify this for me?