Math 6210 • Real Analysis

\begin{enumerate}

\item
State the Baire Category Theorem.


\item
Let \(V\) be a Banach space, and \(T : V \to V\) a bounded linear map.
Assume that for every \(v \in V\) there exists a non-negative integer
\(n\) such that \(T^{n} (v) = 0\). Prove that there exists an integer \(n\) such
that \(T^{n} (v) = 0\) for all \(v \in V\).

\end{enumerate}