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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2020 • Question 4

Question

Math 6210 • Real Analysis Spring 2020 • Question 4

Let \(H\) be a Hilbert space and let \(e_1,e_2,\ldots \) be an orthonormal basis. Prove that \(0\) is a weak limit of the sequence \(e_n\). Prove that the unit ball \(B\), \(\lVert x\rVert \le 1\), is closed in the weak topology. Prove that the unit sphere \(S\), \(\lVert x\rVert =1\), is dense in \(B\) in the weak topology.

Raw LaTeX

Let \(H\) be a Hilbert space and let \(e_1,e_2,\ldots\) be an
orthonormal basis. Prove that \(0\) is a weak limit of the sequence \(e_n\).
Prove that the unit ball \(B\), \(\lVert x\rVert\le1\), is closed in the weak
topology. Prove that the unit sphere \(S\), \(\lVert x\rVert=1\), is dense in
\(B\) in the weak topology.

Topic Tags

Compactness in Metric and Normed Spaces Hilbert Spaces and Orthogonality Lp Spaces and Sequence Spaces Weak Topologies and Weak Convergence