Parameter dependence of solutions of partial differential equations in spaces of real analytic functions

Let $\Omega \subseteq \mathbb{R}^n$ be an open set and let $\mbox{A$(\Omega)$ }$ denote the class of real analytic functions on $\Omega$. It is proved that for every surjective linear partial differential operator $P(D,x):\mbox{A$(\Omega)$ }\to\mbox{A$(\Omega)$ }$ and every family $(f_ \lambda)\subseteq \mbox{A$(\Omega)$ }$depending holomorphically on $\lambda\in \mathbb{C}^m$ there is a solution family $(u_ \lambda)\subseteq\mbox{A$(\Omega)$ }$ depending on $\lambda$ in the same way such that The result is a consequence of a characterization of Fréchet spaces $E$such that the class of ``weakly'' real analytic $E$-valued functions coincides with the analogous class defined via Taylor series. An example shows that the analogous assertions need not be valid if $\mathbb{C}^m$ is replaced by another set.