Weak holomorphic extension of Fréchet-valued functions form compact subsets of \(\mathbb{C}^n\) (Q1775360)

Let \(E, F\) be locally convex spaces and \(X\) be a subset of \(E\). \(f\) is holomorphic on \(X\) if \(f\) can be extended to a holomorphic function \(\widehat f: U \to F\), where \(U\) is a neighborhood of \(X\) in \(E\). \(H(X,F)\) is the space of \(F\)-valued holomorphic functions on \(X\). A function \(f: X \to F\) is called weakly holomorphic if for every \(u\) from the topological dual space of \(F\) the function \(uf:X \to {\mathbb C}\) is holomorphic on \(X\). \(H_w(X,F)\) is the space of \(F\)-valued weakly holomorphic functions on \(X\). The main part of the paper is devoted to the establishment of the equivalence between \(H (X,F)\) and \(H_w (X,F)\), where \(X\) is a compact subset of uniqueness in \(\mathbb C^n\). In the last part the author considers \(F\)-valued separately holomorphic functions.