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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2009 • Question 4

Question

Math 6210 • Real Analysis Spring 2009 • Question 4

Let \(H\) be a Hilbert space and suppose that \(\{v_n\}\) is a sequence in \(H\) with the following property. For each \(u\in H\) \[ \sup _n |\langle v_n,u\rangle |<\infty \] Prove that \(\sup _n\lVert v_n\rVert <\infty \).

Raw LaTeX

Let \(H\) be a Hilbert space and suppose that \(\{v_n\}\) is a
sequence in \(H\) with the following property. For each \(u\in H\)
\[
\sup_n |\langle v_n,u\rangle|<\infty
\]
Prove that \(\sup_n\lVert v_n\rVert<\infty\).

Topic Tags

Hilbert Spaces and Orthogonality Lp Spaces and Sequence Spaces