Functional Analysis – Weak Convergence and Norm Convergence Implying Strong Convergence in Hilbert Space

Question

Let $(x_n)$ be a weakly convergent sequence in a Hilbert space $H$. If $\| x_n \| \to \| x \|$, show that $x_n$ converges strongly to $x$.

Context

This problem comes from a question in my exam paper; the original problem was incorrect.

Answer 1

The result you want to show should be: if $x_n$ converges to $x$ weakly and $\lVert x_n\rVert\to \lVert x\rVert$, then there is convergence in norm. To see that, expand $\lVert x_n-x\rVert^2$ and use the fact that $\langle x_n,x\rangle\to \lVert x\rVert^2$.