# ODE properties true in finite dimension but not in Banach spaces of infinite dimension

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?

• I'd say, those properties that use compactness and not only completeness, like the two you mentioned. A third one should be the statement :" Uniqueness implies continuous dependence", which should have a counter-example in the same space $c_0$. – Pietro Majer Jan 11 '15 at 12:05
• there is a nice statement by de giorgi about long-time behaviour of gradient-like systems. i can give you a precise statement if you are interested in this kind of properties. – Delio Mugnolo Jan 11 '15 at 15:50
• Yes Delio, I'm interested in the statement you mention. Thanks. – mathcounterexamples.net Jan 11 '15 at 18:03