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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Spring 2022 • Question 5

Question

Math 6710 • Applied Linear Operators and Spectral Methods Spring 2022 • Question 5

Let \(X\) be a normed vector space. Let \(x_0 \in X\) and consider the closed ball \(\widetilde {B}(0,\|x_0\|)\). Use the Hahn-Banach theorem to show that there exists \(f \in X'\) such that \(f(x_0) = 1\) and for any \(x \in \widetilde {B}(0,\|x_0\|)\) we have \(f(x) \le 1\). Do not use Hahn-Banach for convex sets or the separating hyperplane theorem.

Raw LaTeX

Let \(X\) be a normed vector space. Let \(x_0 \in X\) and consider the closed ball \(\widetilde{B}(0,\|x_0\|)\).
Use the Hahn-Banach theorem to show that there exists \(f \in X'\) such that \(f(x_0) = 1\)
and for any \(x \in \widetilde{B}(0,\|x_0\|)\) we have \(f(x) \le 1\). Do not use Hahn-Banach for convex
sets or the separating hyperplane theorem.

Topic Tags

Banach and Normed Spaces Convex Sets and Projection Theorems Hahn-Banach Theorem Separation Theorems and Supporting Hyperplanes