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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2019 • Question 3

Question

Math 6210 • Real Analysis Fall 2019 • Question 3

Let \(T:U\to V\) be a linear map between two Banach spaces such that, for any linear functional \(f\) on \(V\), the composite \(f\circ T\) is a continuous functional on \(U\). Prove that \(T\) is continuous. Hint: use the closed graph theorem.

Raw LaTeX

Let \(T:U\to V\) be a linear map between two Banach spaces such
that, for any linear functional \(f\) on \(V\), the composite \(f\circ T\) is a
continuous functional on \(U\). Prove that \(T\) is continuous. Hint: use the
closed graph theorem.

Topic Tags

Banach Spaces Open Mapping and Closed Graph Theorems