# 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.

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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 at 6:37
@AsafKaragila: I changed the perspective. –  András Bátkai Apr 18 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 at 9:25

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.

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Dieudonné's example is beautifully simple! –  André Henriques Apr 18 at 5:17