Corollary 1

If H = E and J ≠ I then f(t, u(t)) is not bounded on J, if, further, E is finite dimensional, then $\|u(t)\|$ has left limit $+\infty$ at the right-hand endpoint of J.

The first part is an immediate consequence of the third part of th. 2 If E is finite dimensional then every closed ball S ⊂ E is compact, so the second part of th. 2 shows that there exists a γ ∈ J such that u(t) ∉ S for t > γ.

If E is finite dimensional it can happen that J ≠ I but that $\|u(t)\|$ remains bounded as t tends to the right-hand endpoint of J (IV, p. 200, exerc. 5).