We propose a new boson expansion theory that reveals and solves the problems of the conventional boson expansion methods that have tried to elucidate nuclear collective motion. The limitation of the number of phonon excitations in addition to phonon excitation modes results in proper boson expansions and allows the use of ideal boson state vectors. The norm operator plays a crucial role and makes it possible to perform boson expansions without the closedalgebra approximation. The treatment of the norm operator determines whether the type of mapping is Hermitian or not, while the boson expansions become infinite expansions regardless of the types of mapping. The closed-algebra approximation makes small parameter expansions impossible and the ideal boson state vectors, having no effect of the Pauli exclusion principle, contain spurious components.