Suppose $H$ is a Hilbert space, and we have a sequence $f_k \rightharpoonup f$ in $H^*$ (the sequence in $H^*$ converges weakly in $H^*$). This implies that $f_k \rightharpoonup^* f$ in $H^*$ (weak-star convergence).
I have a question. Is it possible to commute limit and suprema like this:
$$\lim_{k \to \infty}\sup_{w \in H, \\\lVert w \rVert = 1} \langle f_k, w \rangle_{H^*, H} = \sup_{w \in H\\ \lVert w \rVert = 1} \lim_{k \to \infty} \langle f_k, w \rangle_{H^*, H}$$
Usually I would immediately say no in general, but we do have a weak-star convergence, and I read that this is much nicer than the usual one, so I wonder if it holds?