TITLE: ODE properties true in finite dimension but not in Banach spaces of infinite dimension
QUESTION [14 upvotes]: Some properties of Ordinary Differential Equations - ODE are true in finite dimension spaces but not in Banach spaces of infinite dimension.
The first one I know is the Peano existence theorem. I give a counterexample here for infinite dimension.
The second one states that is the maximal solution of a differential equation is defined in an interval smaller than the one of definition of the map of the unique value problem, then the solution is "exploding". I give a counterexample here for infinite dimension.
Both are from the mathematician Jean Dieudonné.
Do you know other ODE properties valid in finite dimension spaces but not in Banach spaces of infinite dimension?

REPLY [6 votes]: A nice survey is the paper of Lobanov and Smolyanov:
Sergey Grigorievich Lobanov and Oleg Georgievich Smolyanov, Ordinary differential equations in locally convex spaces. Russian Mathematical Surveys 49.3 (1994): 97–175.
The paper lists several counterexamples in the infinite dimensional setting for classical properties and theorems of ODEs in finite-dimensional spaces, such as Peano's theorem, Kneser's theorem, continuous dependence on initial data, continuation of solutions and Picard's theorem.