= Solution
The <Radon-Riesz property> of a norm is the implication
$$
u_k\rightharpoonup u,\quad\|u_k\|\to\|u\|
\quad\Longrightarrow\quad\|u_k-u\|\to0.
$$
In a <Hilbert space>, expand the square using its <inner product>:
$$
\|u_k-u\|^2=\|u_k\|^2+\|u\|^2-2\operatorname{Re}\langle u_k,u\rangle.
$$
<Weak convergence> gives $\langle u_k,u\rangle\to\|u\|^2$, and convergence of norms gives $\|u_k\|^2\to\|u\|^2$. Hence the right side tends to zero. This proves the Hilbert-space <Radon-Riesz theorem>.
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