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Several theorems regarding differential equations are true not only in finite dimension but also in infinite dimension Banach spaces. This is in particular the case for the Cauchy–Lipschitz theorem.

On the other end the Peano existence theorem is false for Banach space with infinite dimensions. See here for a counterexample.

Do you know other counterexamples in "classical" Banach spaces that are different from $c_0$ (the space of sequences of reals converging to $0$)? In particular, is there "an easy example" in the space $C([0,1],\mathbb{R})$ with $\sup$ norm?

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Here is an example in $C([-1,1],R)$, which is a continuous analogue to the discrete example you pointed to: $$ {du(t,x)\over dt} = \operatorname{sign}(u(t,x))\sqrt{|u(t,x)|} + x\;,\qquad u(0,x) = 0\;. $$ For any $t > 0$, the solution (in $L^\infty$) develops a discontinuity at the origin, so that it doesn't belong to $C([-1,1],R)$.

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  • $\begingroup$ Hello Martin, are you sure of your statement? Because for $x=0$ the equation has 2 solutions. The first one is the constant function equal to $0$ and the second one, the function equal to $0$ for $t < 0$ and to $\frac{t^2}{4}$ for $t \ge 0$. The solution of the differential equation seems continuous if you pick-up the second solution for $x=0$. $\endgroup$
    – mathcounterexamples.net
    Commented Aug 31, 2014 at 19:25
  • $\begingroup$ That makes it continuous to the right, but not to the left... $\endgroup$
    – Martin Hairer
    Commented Aug 31, 2014 at 19:28

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