Cone Metric Spaces

Cone metric spaces were introduced in [1] by means of partially ordering real Banach spaces by specified cones. In [4] and [8] , the nation of cone – normed spaces was introduced. cone- metric spaces, and hence, cone- normed spaces were shown to be first countable topological spaces. The reader may consult [5] for this development. In [6], it was shown that, in a sense, cone- metric spaces are not, really, generalizations of metric spaces. This was the motive to do further investigations. Now, we put things in order. 1. Definition:[1] Let (E ,‖∙‖) be a real Banach space and P a subset of E then P is called a cone if : (a) P is closed, convex, nonempty, and P ≠ {0}. (b) a,b ∈ ℝ ; a,b ≥ 0 ; x, y ∈ P ⇒ ax+by ∈ P. (c) x ∈ P and –x ∈ P ⇒ x = 0. 2. Example: [13] Let E= ℓ¹, the absolutely summable real sequences. Then the set P = {x ∈ E : xn ≥ 0 , n} is a cone in E. In our project, we will attempt to enforce the feeling that cone metric spaces are not real generalization of metric spaces by the necessary theory and examples. In the meantime, we will keep it conceivable to arrive at generalization aspects.