I would like to have a counter example that Peano's theorem does not apply to spaces with infinite dimension 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.
 A: Though the answer of Henry Cohn is great, let me mention that Peano's theorem fails in a much broader sense as well, as shown in 
B. M. Garay, Deleting Homeomorphisms and the Failure of Peano's Existence Theorem in Infinite-Dimensional Banach Spaces, Funkcialaj Ekvacioj, 34 (1991), 85--93
A: 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.
A: Since your questions refers to infinite dimensional spaces and not specifically to Banach spaces, it might interest you that there are positive results for locally convex spaces.  These all to a certain extent rely on the fact that in general lcs's, in contrast to Banach spaces, (for example, in Montel spaces or reflexive Banach spaces with the weak topology), one can have large compact sets.  The main principle behind the results is that in the presence of compactness one can employ  the finite dimensional case by reducing to projective limits of spectra of finite dimensional spaces.  One possible reference is the Studia Math. paper of K. Astala "On Peano's theorem in locally convex spaces, 73 (1982), 214-223.
