Problem 1
Given a topological linear space \(E\), suppose \(E\) has sufficiently many continuous linear functionals. Prove that \(\mathrm{E}\) is a Hausdorff space, when equipped with the weak topology.
Problem 2
Let \(\left\\{x_{n}\right\\}\) be a sequence of elements in a Hilbert space \(\mathrm{H}\) such that 1) \(\\{x\), ) converges weakly to an element \(x \in \mathrm{H}\); 2) \(\left\|x_{n}\right\| \rightarrow\|x\|\) as \(n \rightarrow \infty\) Prove that \(\left\\{x_{n}\right\\}\) converges strongly to \(x\), i.e., \(\left\|x_{n}-x\right\| \rightarrow 0\) as \(n \rightarrow \infty\),
Problem 4
Let \(H\) be a (separable) Hilbert space and M a bounded subset of \(\mathrm{H}\). Prove that the topology in \(\mathrm{M}\) induced by the weak topology in \(\mathrm{H}\) can be specified by a metric.
Problem 5
Prove that every closed convex subset of a Hilbert space \(\mathrm{H}\) is closed in the weak topology (so that, in particular, every closed linear subspace of \(\mathrm{H}\) is weakly closed). Give an example of a closed set in \(\mathrm{H}\) which is not weakly closed.
