Today in an exam on functional analysis the following question was posed:
Let $H$ be a Hilbert space and $(x_n)_{n\in \Bbb{N}} \subseteq H$ be a sequence that converges weakly to $x\in H$ satisfying $\|x_n\|\to \|x\|$ as $n\to \infty$. Show that $(x_n)_{n\in \Bbb{N}}$ converges strongly to $x$.
I am aware of a proof based writing the norm with the scalar product.
I was wondering whether this also holds under weaker assumptions and found this answer. My question basically is whether we can better classify where norm convergence or the $\limsup$ estimate in the quoted answer is sufficient or in the best case an equivalent description.