Factorization of uniformly holomorphic functions

Let E be a complex Hausdorff locally convex space such that the strong dual E' of E is sequentially complete, let F be a closed linear subspace of E and let U be a uniformly open subset of E. We denote by Π: E → E/F the canonical quotient mapping. In §1 we study the factorization of uniformly holomorphic functions through π. In §2 we study F-quotients of uniform type and introduce the concept of envelope of uF-holomorphy of a connected uniformly open subset U of E. The main result states that the pull-back ${\displaystyle {\epsilon }_u^{\ast }\left(U\right)}$ of the envelope of uniform holomorphy of Π(U) constructed by Paques and Zaine [9] is the envelope of uF-holomorphy of U.