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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 \(H\) be a Hilbert space with inner product \((\cdot ,\cdot )\). Suppose \(\{x_n\}\) is a sequence of elements in \(H\) with the property that the sequence \(\{(x_n,y)\}\) converges for each \(y\in H\). Prove that there is an element \(x\in H\) such that \(\{(x_n,y)\}\) converges to \((x,y)\) for each \(y\in H\).

Raw LaTeX

Let \(H\) be a Hilbert space with inner product \((\cdot,\cdot)\).
Suppose \(\{x_n\}\) is a sequence of elements in \(H\) with the property that the
sequence \(\{(x_n,y)\}\) converges for each \(y\in H\). Prove that there is an
element \(x\in H\) such that \(\{(x_n,y)\}\) converges to \((x,y)\) for each
\(y\in H\).

Topic Tags

Hilbert Spaces and Orthogonality Lp Spaces and Sequence Spaces