Radon-Riesz theorem

In a Hilbert space, weak convergence $x_n\rightharpoonup x$ together with $\Vert x_n\Vert \to \Vert x\Vert$ implies strong convergence $x_n\to x$.

Suppose $x_n\rightharpoonup x$ in a separable Hilbert space with Hilbertian basis $(e_i)$. Then $x_n\to x$ in norm exactly whenThe condition prevents norm from escaping to successively higher basis coordinates.