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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2019 • Question 3

Question

Math 6210 • Real Analysis Spring 2019 • Question 3

Let \(X\) be a normed space, and let \(X^*\) be the space of bounded linear functionals on \(X\). For every \(x\in X\), let \(x^*\) be the functional on \(X^*\) defined by \(x^*(y)=y(x)\) for all \(y\in X^*\). Prove that \(x\mapsto x^*\) is an isometry from \(X\) into \(X^{**}\).

Raw LaTeX

Let \(X\) be a normed space, and let \(X^*\) be the space of bounded
linear functionals on \(X\). For every \(x\in X\), let \(x^*\) be the functional on
\(X^*\) defined by \(x^*(y)=y(x)\) for all \(y\in X^*\). Prove that \(x\mapsto x^*\)
is an isometry from \(X\) into \(X^{**}\).

Topic Tags

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