Power of a bounded linear map nullify each point implies the power is eventually zero

Question

I am working on a problem from my Qual

"Let $T:V\to V$ be a bounded linear map where $V$ is a Banach space. Assume for each $v\in V$, there exists $n$ s.t. $T^n(v)=0$. Prove that $T^n=0$ for some $n$."

My impression is that it looks like a algebraic problem. But I know nothing about $V$ (not fintely generated or something like that). So this one is off.

The hypothesis seems to be set up for the Uniformly Bounded Principle. Indeed, for each $v$, $\sup_n \{||T^n(v)|| \}$ is bounded, so by the principle, $\sup_n ||T^n||$ is bounded.

I'm stuck here. I don't think there is a relation between $||T^n||$ and $||T||$, even if there is, I cannot shake the feeling that the sequence $||T^n||$ may be eventually constant. I am hoping $||T^n||=0$ eventually but failed to prove it.

Answer 1

Your hypothesis gives you that $V=\bigcup_n\ker T^n$, a union of closed subspaces. By Baire's Category Theorem, one of them contains a ball, and so it's equal to $V$.

Answer 2

Try writing $X$ as a countable union of closed sets, and concluding that one of then contains a neighborhood.