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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2010 • Question 3

Question

Math 6210 • Real Analysis Fall 2010 • Question 3

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

Raw LaTeX

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

Topic Tags

Hilbert Spaces and Orthogonality Lp Spaces and Sequence Spaces