  We propose a new boson expansion theory that does not premise the
closed-algebra approximation, indispensable for formulation until now, as an
extension of the conventional practical boson expansion methods that have tried
to elucidate nuclear collective motion, which is a method that allows the
closed-algebra approximation not to be used or to be used appropriately,
enables us to obtain the boson expansion easier, and reproduces the fermion
subspace onto the boson subspace more faithfully than the conventional
practical methods. The two-phonon norm matrix composed of all the phonon
excitation modes is investigated in detail, which reveals the mechanism,
essential for the boson expansion methods, of how we should construct the
fermion subspace to be mapped from the whole fermion space. The conventional
practical boson expansion methods have applied the closed-algebra approximation
improperly to strengthen the effect of the Pauli principle inappropriately,
which should be replaced by those that do not use that approximation or use it
properly.
