The key result (Theorem 1) provides the existence of a holomorphic approximation map for some space of C∞-functions on an open subset of Rn. This leads to results about the existence of a continuous linear extension map from the space of the Whitney jets on a closed subset F of Rn into a space of holomorphic functions on an open subset D of Cn such that D ∩ Rn = RnF.

This article is devoted to a study of locally convex topologies on $\mathbf{H}\left(U\right)$ (where $U$ is an open subset of the locally convex topological vector space $E$ and $\mathbf{H}\left(U\right)$ is the set of all complex valued holomorphic functions on $E$). We discuss the following topologies on $\mathbf{H}\left(U\right)$:(a) the compact open topology ${\mathbf{I}}_0$,(b) the bornological topology associated with ${\mathbf{I}}_0$,(c) the ported topology of Nachbin ${\mathbf{I}}_{\omega }$,(d) the bornological topology associated with ${\mathbf{I}}_{\omega }$ ; and(e) the ${\mathbf{I}}_{\omega }$ topological of Nachbin.For $U$ balanced we show these topologies are...

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...