One of the generalizations that were studied from metric space was multiplicative metric space. The main idea was that the usual triangular inequality was replaced by a multiplicative triangle inequality. The important thing is that logarithm of every multiplicative metric is a metric. In this paper, we introduce multiplicative norm space and present three norms in bounded multiplicative operator spaces and we investigate conditions that bounded multiplicative operator spaces be complete norm multiplicative spaces. It is notable that the logarithm of every multiplicative norm is not a norm and so we have new results in multiplicative norm spaces. We give an important extension of the Hahn-Banach theorem to nonlinear operators and their ramifications and indicate some applications.