Let {fm} m≥1 be a sequence of holomorphic functions defined on a bounded domain D ⊂ C n or a sequence of rational functions (1 ≤ deg rm ≤ m) defined on C n . We are interested in finding sufficient conditions to ensure the convergence of {fm} m≥1 on a large set provided the convergence holds pointwise on a not too small set. This type of result is inspired from a theorem of Vitali which gives a positive answer for uniformly bounded sequence.

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We establish the equivalence between the weak holomorphicity and holomorphicity of Fréchet-valued functions on compact polydisks in (DFN)-spaces. Moreover, the relations between separately holomorphic functions and holomorphic functions on compact polydisks in (DFN)-spaces are also given.