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

Let Omega subset of or equal to R(n) be an open set and let 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) :A(Omega) -->A(Omega) and every family (f(lambda)) subset of or equal to A(Omega) depending holomorphically on lambda is an element of C(m) there is a solution family (f(lambda)) subset of or equal to A(Omega) depending on lambda in the same way such that P(D;x)u(lambda) = f(lambda); for lambda is an element of C(m). The result is a consequence of a characterization of Frechet 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 C(m) is replaced by another set.