On the uniformity of meromorphic functions (Q1966091)

Let \(E\) and \(F\) be locally convex spaces and \(D\subset E\) open. An \(F\)-valued function \(f\) is said to be meromorphic on \(D\) if it is holomorphic on a dense open subset and is locally the quotient of two holomorphic functions. It is said to be of uniform type if it factors meromorphically through a Banach space. The author gives conditions, in terms of the linear topological invariants, for all meromorphic functions to be of uniform type (\(M=M_u\)): a) If \(E\) is nuclear Fréchet, then \(M(E',F')=M_u(E',F')\) for every Fréchet space \(F\in (LB^\infty)\) if and only if \(E\in (DN)\). b) If \(F\) is Fréchet, then \(M(E',F')=M_u(E',F')\) for every nuclear Fréchet space \(E\in (DN)\) if and only if \(F\in (LB^\infty)\). c) If \(E\) is a Fréchet-Montel space with property \((DN)\) and \(F\) a Fréchet space with property \((\overline{\overline{\Omega}})\), then \(M(E',F')=M_u(E',F')\).