Spaces of holomorphic functions in regular domains

Let Ω be a regular domain in the complex plane C , Ω ≠ C . Let G b ( Ω ) be the linear space over C of the holomorphic functions f in Ω such that f ( n ) is bounded in Ω and is continuously extendible to the closure Ω ¯ of Ω, n = 0 , 1 , 2 , …  . We endow G b ( Ω ) , in a natural manner, with a structure of Fréchet space and we obtain dense subspaces F of G b ( Ω ) , with good topological linear properties, also satisfying that each function f of F, distinct from zero, does not extend holomorphically outside Ω.