Some results on solvability of ordinary linear differential equations in locally convex spaces

Let $\Gamma$ be the class of sequentially complete locally convex spaces such that an existence theorem holds for the linear Cauchy problem $\dot x=Ax$, $x(0)=x_0$, with respect to functions $x\colon\mathbf R\to E$. It is proved that if $E\in\Gamma$, then $E\times\mathbf R^A\in\Gamma$ for an arbitrary set $A$. It is also proved that a topological product of infinitely many infinite-dimensional Fréchet spaces, each not isomorphic to $\mathbf R^\infty$, does not belong to $\Gamma$.