  In this paper we prove a variation of the theorem in title, for equations
with periodic coefficients, in Frechet spaces. The main result gives equivalent
conditions ensuring the reduction of such an equation to one with constant
coefficient. In the particular case of $C^{\infty}$, we obtain the exact
analogue of the classical theorem. Our approach essentially uses the fact that
a Frechet space is the limit of a projective sequence of Banach spaces,
consequently a linear differential equation in the former can be obtained as
the projective limit of a system of linear equations in the latter (see also
[G. Galanis, On a type of linear differential equations in Frechet spaces, Ann.
Scuola Norm. Sup. Pisa, Ser. 4, 24 (1997), 501-510]). A geometric application
is also discussed.
