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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Fall 2022 • Question 4

Question

Math 6710 • Applied Linear Operators and Spectral Methods 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

\[\|x\| = \sup _{f \in X', f \neq 0} \frac {|f(x)|}{\|f\|}.\]

Raw LaTeX

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

\[\|x\| = \sup_{f \in X', f \neq 0} \frac{|f(x)|}{\|f\|}.\]

Topic Tags

Banach and Normed Spaces Hahn-Banach Theorem