  G\"ahler ([4],[5]) introduced and investigated the notion of 2-metric spaces
and 2-normed spaces in sixties. These concepts are inspired by the notion of
area in two dimensional Euclidean space. In this paper, we choose a
fundamentally different approach and introduce a possible generalization of
usual norm retaining the distance analogue properties. This generalized norm
will be called as $G$-norm. We show that every $G$-normed space is a $G$-metric
space and therefore, a topological space and develop the theory for $G$-normed
spaces. We also introduce $G$-Banach spaces and obtain some fixed point
theorems.
