Fréchet-valued real analytic functions on Fréchet spaces (Q1408504)

Let \(E\) be a real Fréchet space, \(D\subset E\) open, \(F\) a complex Fréchet space and \(f:D\to F\) a function. Then \(f\) is called topologically real analytic if locally \(f\) admits a power series expansion, while it is called real analytic if for each \(u\in F'\) the function \(u\circ D\to \mathbb{C}\) is topologically real analytic. Let \(A_t(D,F)\) (resp. \(A(D,F)\)) denote the space of all topologically real analytic (resp. real analytic) functions on \(D\) with values in \(F\). The authors prove Theorem A. Let \(F\) be a Fréchet space. Then the following are equivalent: (i) \(F\) has property \((DN)\); (ii) \(A(E,F)= A_t(E,F)\) for every real nuclear Fréchet space \(E\) having property \((\widetilde{\Omega})\); (iii) \(A(E,F)= A_t(E,F)\) for every real Fréchet-Schwartz space \(E\) having property \((\widetilde{\Omega})\) and an absolute basis \(\{e_j\}_{j\geq 1}\). Theorem B. Let \(F\) be a Fréchet space having property \((LB_\infty)\), then \(A(E,F)= A_t(E,F)\) holds for all real Fréchet spaces \(E\). Note that Theorem A extends a result of \textit{J. Bonet} and \textit{P. Domanski} [Monatsh. Math. 126, 13--26 (1998; Zbl 0918.46034)] for \(E=\mathbb{R}\).