I'm looking for a rigorous mathematical description of the quantum mechanical space state of, for instance, a particle with no internal states. At university we were told that it the Hilbert state of wave functions. They gave us no particular restrictions on these functions, such as continuity, apart from the fact that they should be quadratically integrable, so that we can define inner products. But then we are given a basis of kets of the form ##| \vec r>##, representing "Dirac functions", that precisely are not quadratically integrable. We were told to shut up and calculate. I'd like to clear this up a bit. I know the Dirac "functions" can be defined rigorously as distributions. But how does that fit into a Hilbert space? Does someone know of a mathematically rigorous, but elementary, introduction to this issue?