Math 6210 • Real Analysis

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.

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\textbf{Section B.}
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\textbf{Notation:} \(D=\{z\in\mathbb C:|z|<1\}\).