Fréchet-valued meromorphic functions on compact sets in \(\mathbb{C}^n\) (Q1281599)

Let \(X\) be denoted a subset of \(\mathbb{C}^n\), \(F\) a sequentially complete locally convex space, \(M(X,F)\) the vector space of meromorphic functions on \(X\) with values in \(F\) (that is a function which can be extended to a meromorphic function on a neighbourhood of \(X\) in \(\mathbb{C}^n\)) and \(M_w(X,F)\) be the vector space of weakly meromorphic functions (that is a function \(f\) such that \(x^*f\) is a mermorphic function, for every \(x^*\in F^*\), \(F^*\) the dual space of \(F\)) on \(X\) with values in \(F\). The purpose of this paper is to find necessary and sufficient conditions for which \(M(X,F)= M_w(X,F)\). Two such conditions are given: Theorem A. Let \(F\) be a Fréchet space. Then \([F_{\text{bor}}^*]^*\in (LB_\infty)\) if and only if \(M(X, [F_{\text{bor}}^*]^*)=M_w(x,[F^*_{\text{bor}}]^*)\) for every compact uniqueness set \(X\) of \(\mathbb{C}^n\). Here \(F_{\text{bor}}^*\) denotes the space \(F^*\) equipped with the bornological topology. Theorem B. Let \(X\) be a compact subset of \(\mathbb{C}^n\). The following are equivalent: (i) \(X\) is not pluripolar; (ii) \([{\mathcal H}(X)']\in (LB^\infty)\); (iii) \(X\) is unique and \(M(X,F)= M_w(X,F)\) for every Fréchet space \(F\) with \(F\in (DN)\).