Yes, and I think that in introductory qm courses it is quite OK to skip this.
I come from a mathematical physics community and the usual terminology over there is:
A is hermitian = bounded and A = A* (where A* in this case is the adjoint of A, which is defined as a bounded operator using...
Well, I would disagree. It is not only matter of a mathematical rigor.
In this discussion there was no mention of the problem of finding an actual self adjoint operator. That is, one usualy starts with some symmetric (some authors call it hermitian, even it is not everywhere defined) operator...