Topic
Hahn-Banach Theorem
Courses
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Questions
This topic is tagged for 4 questions
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Fall 2024 • Question 5
Let \(X\) be a normed space over \(\mathbb {R}\) and \(Y \subset X\) be a linear subspace. For \(x_0 \in X \setminus Y\) we define the linear functional
\[f : \operatorname {span}\{Y, x_0\} \to \mathbb {R}\]by
\[f(y + \lambda x_0) = \lambda , \quad \text {for all } y \in Y, \lambda \in \mathbb {R}.\]- Prove that \(f\) is continuous if and only if \(x_0 \notin \overline {Y}\).
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Prove that \(\overline {Y} = \bigcap \{\operatorname {Ker} g \mid g \in X', Y \subset \operatorname {Ker} g\}\), where \(X'\) is the dual of \(X\).
Hint for part \((ii)\): The \(\subset \) direction is relatively easy. For the other direction, note that as given in part (i), \(f\) satisfies \(f|_Y = 0\) and \(f(x_0) = 1\). The Hahn-Banach theorem provides an extension of \(f\) to the entire space, use this to prove the other direction by contradiction.
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Spring 2024 • Question 3
Let \(X\) be a normed vector space. Use the Hahn-Banach theorem to show that for any \(x \in X\) we have that
\[\|x\| = \sup _{f \in X', f \neq 0} \frac {|f(x)|}{\|f\|}.\] -
Fall 2022 • Question 4
Let \(X\) be a non-trivial normed vector space. Use the Hahn-Banach theorem to show that for any \(x \in X\) we have that
\[\|x\| = \sup _{f \in X', f \neq 0} \frac {|f(x)|}{\|f\|}.\] -
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.