Abstract
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 Frechet spaces E (resp. \({\mathbb{C}^n}\)) where \({W \subset F'}\) defines the topology of Frechet 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 Frechet-valued holomorphic functions are also proved.
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