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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2009 • Question 3

Question

Math 6210 • Real Analysis Fall 2009 • Question 3

Let \(H\) be a Hilbert Space, and suppose that \(\{x_n\}\) is a sequence in \(H\) such that \(\langle x_n,y\rangle \) is convergent for each \(y\in H\). Prove that there exists \(x\in H\) such that \(\langle x_n,y\rangle \to \langle x,y\rangle \) for all \(y\in H\).

Raw LaTeX

Let \(H\) be a Hilbert Space, and suppose that \(\{x_n\}\) is a
sequence in \(H\) such that \(\langle x_n,y\rangle\) is convergent for each
\(y\in H\). Prove that there exists \(x\in H\) such that
\(\langle x_n,y\rangle\to\langle x,y\rangle\) for all \(y\in H\).

Topic Tags

Hilbert Spaces and Orthogonality Lp Spaces and Sequence Spaces