The space H(Ω,(zj)) of holomorphic functions

Let Ω be a domain in Cn. Let H(Ω) be the linear space over C of the holomorphic functions in Ω, endowed with the compact-open topology. Let (zj) be a sequence in Ω without adherent points in Ω. In this paper, we define the space H(Ω,(zj)) and some of its linear topological properties are studied. We also show that, for some domains of holomorphy Ω and some sequences (zj), the non-zero elements of H(Ω,(zj)) cannot be extended holomorphically outside Ω. As a consequence, we obtain some characterizations of the domains of holomorphy in Cn.