Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2004 • Question 5

Question

Math 6210 • Real Analysis 2004 • Question 5

Let \(\lambda \) be a bounded linear functional on a linear subspace \(V\) of a Hilbert space \(H\). Show that there exists a unique bounded linear functional \(\Lambda \) on \(H\) such that: (a) \(\|\Lambda \|=\|\lambda \|\), and (b) \(\Lambda (x)=\lambda (x)\) for \(x\in V\).

Raw LaTeX

Let \(\lambda\) be a bounded linear functional on a linear subspace
\(V\) of a Hilbert space \(H\). Show that there exists a unique bounded linear
functional \(\Lambda\) on \(H\) such that: (a) \(\|\Lambda\|=\|\lambda\|\), and
(b) \(\Lambda(x)=\lambda(x)\) for \(x\in V\).

Topic Tags

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