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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2005 • Question 3

Question

Math 6210 • Real Analysis 2005 • Question 3

Let \(f\) be a bounded linear functional on a linear subspace \(M\) of a Hilbert space \(H\).

(i) Prove that \(f\) has a unique extension to a bounded linear functional \(F\) on \(H\) so that the norm of \(f\) coincides with the norm of \(F\).

(ii) Prove that the extension \(F\) vanishes on \(M^\perp \).

Raw LaTeX

Let \(f\) be a bounded linear functional on a linear subspace \(M\) of a
Hilbert space \(H\).

(i) Prove that \(f\) has a unique extension to a bounded linear functional \(F\) on
\(H\) so that the norm of \(f\) coincides with the norm of \(F\).

(ii) Prove that the extension \(F\) vanishes on \(M^\perp\).

Topic Tags

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