A sequences $\{h_n\}$ in a Hilbert space $\mathcal{H}$ is said to converge weakly to $h \in \mathcal{H}$ if
$$ \lim_{n \to \infty} \langle h_n, g \rangle = \langle h, g \rangle $$
for every $g \in \mathcal{H}$.
Show that if $h_n \to h$ in norm, then $h_n \to h$ weakly. Show that the converse is false, but that if $h_n \to h$ weakly and $||h_n|| \to ||h||$, then $h_n \to h$ in norm.
