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# Question on Linear Operators

Let $V$ be a normed infinite dimensional vector space. Let $L: V \longrightarrow V$ be a bounded linear operator. Moreover assume that $L$ is 'locally nilpotent' that is: $$\forall v \in V \quad \exists n \in \mathbf{N}: L^n (v) = 0.$$ Now my question is if the linear operator: $$\exp (L) = \sum_{n=0}^{\infty} \frac{L^n}{n!}$$ is bounded or not.
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