# 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.

• The title sounds like something a waiter in a mathematical restaurant would ask a patron after ordering "Peano's theorem" and some bitter lemon please. – Asaf Karagila Apr 18 '14 at 6:37
• @AsafKaragila: I changed the perspective. – András Bátkai Apr 18 '14 at 9:17
• @András: So now we're ordering directly. The follow up, if so, is "Would you like fries with that?" ;-) – Asaf Karagila Apr 18 '14 at 9:25