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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

1998 • Question 3

Question

Math 6210 • Real Analysis 1998 • Question 3

Let \(T\) be a bounded linear functional on a (linear) subspace \(M\) of a Hilbert space \(H\). Prove that \(T\) has a unique norm preserving extension to a bounded linear functional on \(H\), and that this extension vanishes on \(M^\perp \).

Raw LaTeX

Let \(T\) be a bounded linear functional on a (linear) subspace \(M\) of
a Hilbert space \(H\). Prove that \(T\) has a unique norm preserving extension to a
bounded linear functional on \(H\), and that this extension vanishes on \(M^\perp\).

Topic Tags

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