  Consider a smooth vector field $f\colon \mathbb{R}^n\to\mathbb{R}^n$ and a
maximal solution $\gamma\colon \,]a,b[\,\to \mathbb{R}^n$ to the ordinary
differential equation $x'=f(x)$. It is a well-known fact that, if $\gamma$ is
bounded, then $\gamma$ is a global solution, i.e., $\,]a,b[\,=\mathbb{R}$. We
show by example that this conclusion becomes invalid if $\mathbb{R}^n$ is
replaced with an infinite-dimensional Banach space.

Functional Analysis