Some Results on TVS-cone Normed Spaces and Algebraic Cone Metric Spaces

In this paper we introduce the cone bounded linear mapping and show that the cone norm is continuous. Among other things, we prove the open mapping theorem and the closed graph theorem in TVS-cone normed spaces. We also show that under some restrictions on the cone, two TVS-cone norms are equivalent if and only if they induce equivalent topologies. In the sequel, the notion of algebraic cone metric is introduced and it is shown that every algebraic cone metric space has a topology and the Banach fixed point theorem for contraction mappings on algebraic cone metric spaces is proved.