  Using unbounded Hilbert space representations basic results on the transition
probability of positive linear functionals $f$ and $g$ on a unital *-algebra
are obtained. The main assumption is the essential self-adjointness of GNS
representations $\pi_f$ and $\pi_g$. Applications to functionals given by
density matrices and by integrals and to vector functionals on the Weyl algebra
are given.
