We introduce a new norm, called the $N^p$-norm (1 $leq$ p < ${infty}$ on the space $N^p$(V,W) where V and W are abstract operator spaces. By proving some fundamental properties of the space $N^p$(V,W), we also discover that if W is complete, then the space $N^p$(V,W) is also a Banach space with respect to this norm for 1 $leq$ p < ${infty}$.