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Yemon Choi
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A nice survey is the paper of Lobanov, and Smolyanov:

Sergey Grigorievich, Lobanov and Oleg Georgievich Smolyanov. ", Ordinary differential equations in locally convex spaces.Ordinary differential equations in locally convex spaces." Russian Mathematical Surveys 49.3 (1994): 97-17597–175.

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.

A nice survey is Lobanov, Sergey Grigorievich, 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.

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.

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Tadashi
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A nice survey is Lobanov, Sergey Grigorievich, 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.