Generalized Normed Spaces and Fixed Point Theorems

G?hler ([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 -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.