ABSTRACT
For a quantum system with observables capable of assuming an infinite number of different values we would need an infinite-dimensional complex vector space to serve as its state space. Every Cauchy sequence of vectors in a finite-dimensional complex vector space converges to a limit vector in the space the same is not true in an arbitrary infinite-dimensional complex scalar product space. Fortunately infinite-dimensional scalar product spaces of physical importance are such that all Cauchy sequences do converge. Such spaces are called Hilbert spaces. Since every Cauchy sequence in a finite-dimensional scalar product space converges they are automatically Hilbert spaces. The situation in infinite-dimensional spaces is different. With Riemann integrals for square integrability and the scalar product replaced by Lebesque integrals we would obtain a larger set of functions which are Lebesque square-integrable.
