On σ(·, W)-Holomorphic Functions and Theorems of Vitali-Type.

The aim of this paper is to give some criterions for holomorphy of F-valued σ( F, W)-holomorphic functions which are bounded on bounded sets in a domain D of Fréchet spaces E (resp. $${\mathbb{C}^n}$$) where $${W \subset F'}$$ defines the topology of Fréchet space F. Base on these results we consider the problem on holomorphic extension of F-valued σ( F, W)-holomorphic functions from non-rare subsets of D and from subsets of D which determines uniform convergence in H( D). As an application of the above, some theorems of Vitali-type for a locally bounded sequence $${\{f_i\}_{i \in \mathbb{N}}}$$ of Fréchet-valued holomorphic functions are also proved. [ABSTRACT FROM AUTHOR]