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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2019 • Question 4

Question

Math 6210 • Real Analysis Fall 2019 • Question 4

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 \(v\in V\).

Raw LaTeX

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 \(v\in V\).

Topic Tags

Banach Spaces Bounded Linear Operators