Weak convergence plus strong convergence

Question

Let $H$ be a Hilbert space and let $(x_n), (y_n)$ be sequences in $H$ such that $(x_n)$ converges strongly to $x$ and $(y_n-x_n)$ converges weakly to 0. I can show that $(y_n)$ converges weakly to $x$, but is it also true that $y_n$ converges strongly to $x$? I've tried to prove it by showing that $||y_n|| \rightarrow ||x||$ strongly, but I can't seem to make the argument work.

Answer 1

As pointed out by Daniel Fischer, it's not true without further assumptions, even in the case $x_n=x=0$. In this case, the assumption reduces to $y_n\to 0$ weakly, while the wanted conclusion is $\lVert y_n\rVert\to 0$. In an infinite dimensional Hilbert space, there always exists weakly convergent sequences which are not norm-convergent.