Dugundji s extension theorem in TVS-cone metric spaces

The Dugundji extension theorem plays an important role in the topological theory. It states that every continuous map from a closed subset A of the metric space X into a locally convex space Y can be continuously extended to X. Note that the well-known Tietze extension theorem is a consequence of the Dugundji extension theorem. Recently, Kieu Phuong Chi and Tran Van An [Kieu Phuong Chi and Tran Van An [K.P. Chi, T. Van An, Dugundji s theorem for cone metric spaces, Appl. Math. Lett. (2010) doi:10.1016/j.aml.2010.10.034] proved Dugundji s extension theorem for the normal cone metric space. The aim of this paper is to prove this in the frame of the TVS - cone spaces in which the cone does not need to be normal.