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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2021 • Question 4

Question

Math 6210 • Real Analysis Spring 2021 • Question 4

  1. State the Closed Graph Theorem.
  2. Let \(T : U \to V\) be a linear map between two Banach spaces such that, for any bounded linear functional \(f\) on \(V\) , the composite \(f \circ T\) is a bounded functional on \(U\) . Prove that \(T\) is bounded. You may use the fact that bounded linear functionals on a Banach space separate points.

Raw LaTeX

\begin{enumerate}

\item
State the Closed Graph Theorem.


\item
Let \(T : U \to V\) be a linear map between two Banach spaces such
that, for any bounded linear functional \(f\) on \(V\) , the composite \(f \circ T\)
is a bounded functional on \(U\) . Prove that \(T\) is bounded. You may
use the fact that bounded linear functionals on a Banach space
separate points.

\end{enumerate}

Topic Tags

Banach Spaces Bounded Linear Operators Dual Spaces and Hahn-Banach Theorem Open Mapping and Closed Graph Theorems