MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
up vote 21 down vote accepted

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,, 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

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.

share|cite|improve this answer
Dieudonné's example is beautifully simple! – André Henriques Apr 18 '14 at 5:17

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.

share|cite|improve this answer

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.