as if one has just a linear scalar product space $\Psi$.

Calculating just in the pre-Hilbert space $\Psi$ — as physicists usually do — the RHS formulation is really not more difficult than the Hilbert space formulation. One just has to use a slightly more general set of rules for these calculations. This has always been done in the Dirac formalism of bra’s and ket’s. In addition to the rules of the Dirac formalism, the RHS provides a mathematical justification for additional rules of mathematical manipulations. The most important of these are:

1.  the eigenvectors of self-adjoint observables $A$ (i.e. with $A^\dagger=\overline{A}$) in (\[3en\]) can be complex

2.  the time evolution for some of the solutions of the Schrödinger equation can be given by a semigroup and not by a reversible unitary group

3.  some vectors can decay exponentially (as envisioned by Gamow).

Dynamical equations (laws of nature) are the same in both the Hilbert space and the RHS formulations, namely given by the Schrödinger equation $$i\hbar {\partial\left|\phi(t)\right\rangle\over\partial t}
  =
  H \left|\phi(t)\right\rangle.$$ or the von Neumann equation (2) with $\delta H=0$. But in the RHS formulation different initial and boundary conditions than in the Hilbert space formulation allow for a 