  We investigate Dirac's bra-ket formalism based on a rigged Hilbert space for
a non-Hermite quantum system with a positive-definite metric. First, the rigged
Hilbert space, characterized by positive-definite metric, is established. With
the aid of the nuclear spectral theorem for the obtained rigged Hilbert space,
spectral expansions are shown for the bra-kets by the generalized eigenvectors
of a quasi-Hermite operator. The spectral expansions are utilized to endow the
complete bi-orthogonal system and the transformation theory between the Hermite
and non-Hermite systems. As an example of application, we show a specific
description of our rigged Hilbert space treatment for some parity-time
symmetrical quantum systems.
