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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2020 • Question 3

Question

Math 6210 • Real Analysis Spring 2020 • Question 3

Let \(X\) be a normed space and \(X^*\) be the space of bounded linear functionals. Every \(x\in X\) defines a functional \(\hat {x}\) on \(X^*\) by \(\hat {x}(f)=f(x)\) for all \(f\in X^*\). Prove that \(\lVert \hat {x}\rVert =\lVert x\rVert \).

Raw LaTeX

Let \(X\) be a normed space and \(X^*\) be the space of bounded
linear functionals. Every \(x\in X\) defines a functional \(\hat{x}\) on \(X^*\) by
\(\hat{x}(f)=f(x)\) for all \(f\in X^*\). Prove that
\(\lVert\hat{x}\rVert=\lVert x\rVert\).

Topic Tags

Banach Spaces Bounded Linear Operators Dual Spaces and Hahn-Banach Theorem