Consider \(L^2(\mathbb Z):=H\). Recall two topologies on the set of bounded linear operators on \(H\): The norm topology which is metric topology given by \[ d(A,B)=\sup _{v\in H:\lVert v\rVert =1}\lVert Av-Bv\rVert . \] The strong operator topology which is given by \(A_i\to A_\infty \) if for every \(v\in H\) we have \[ \lim _{i\to \infty }\lVert A_iv-A_\infty v\rVert =0. \]
(a) Show that if \(A_i\) converges to \(A_\infty \) in the norm topology then it converges to \(A_\infty \) in the strong operator topology.
(b) Give an example of \(A_i\) which converges to \(A_\infty \) in the strong operator topology but not the norm topology.
(c) Let \(A_i\) be the operator defined by \(A_iv=w\) where \(w_j=v_j\) if \(j\ne i\) and \(w_i=iv_i\). Show \(A_1,\ldots \) is a sequence of operators that does not converge in either topology.