In this article we discuss the relationship between domains of existence domains of holomorphy, holomorphically convex domains, pseudo convex domains, in the context of locally convex topological vector spaces. By using the method of Hirschowitz for ${\Pi }_{n=1}^{\infty }\mathbf{C}$ and the method used for Banach spaces with a basis we prove generalisations of the Cartan-Thullen-Oka-Norguet-Bremmerman theorem for finitely polynomially convex domains in a variety of locally convex spaces which include the following:1) $N$-projective...