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?
Best Answer
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 $C([-1,1],R)$.