**Does Peano's theorem apply to spaces with infinite dimension? Or is there a counterexample?**


Here, Peano's theorem is:


Let $E$ be a space with finite dimension. Consider a point $(t\_0,x\_0) \in \Re \times E$, constants $ a, b > $ 0 and a continuous function $$F: [t\_0 - a, t\_0 + a] \times B\_b[x\_0] \longrightarrow E$$
Then for every $ M> $ 0 satisfying
$$\sup \{||F(t,x)||:(t,x) \in [t\_0 - a, t\_0 + a] \times B\_b[x\_0]\} < M$$
the Cauchy problem
$$x'(t)=F(t,x(t));\ \ x(t\_0)=x\_0$$
admits at least one solution in the interval:
$$\big[t\_0 - \min(a,\frac{b}{M}),t\_0 + \min(a,\frac{b}{M})\big] $$
An infinite-dimensional counterexample would be of great help. Thank you very much.



I would like to have a counter example that Peano's theorem does not apply to spaces with infinite dimension
No, Peano's existence theorem fails completely in infinite-dimensional spaces: there are counterexamples in every infinite-dimensional Banach space. This is a theorem of Godunov (A. N. Godunov, *Peano's theorem in Banach spaces*, Functional Analysis and its Applications **9** (1975), 53-55, <http://dx.doi.org/10.1007/BF01078180>), while the first counterexample in some Banach space was due to Dieudonné (J. Dieudonné, *Deux exemples singuliers d'équations différentielles*, Acta Sci. Math. Szeged. **12:B** (1950), 38-40; see <http://acta.fyx.hu/>).


It's not hard to see that the finite-dimensional proof fails (the weak point is generally when Arzelà–Ascoli is applied), but I still find it surprising that the theorem fails as well, since it naively sounds like it should obviously be true.


