PEANO THEOREM IS FALSE FOR ANY INFINITE-DIMENSIONAL FRECHET SPACE

It is proved that for any nonnormable Frechet space E a continuous map f : E --> E and a closed infinite-dimensional subspace L can be found such that the Cauchy problem x = f(x) , x(0) = u has no solution for any u is-an-element-of L. Previous counterexamples to Peano's theorem cover Banach spaces and nonsemireflexive spaces.