Assume $\{x,x_1,x_2,\ldots\}$ is a subset of a given Hilbert space $X$ with norm $||\cdot||$ such that $x_j$ converges to $x$ weakly and $||x_j||$ converges to $||x||$. Prove that $x_j$ converges to $x$ in norm.
For this question, I think that since $X$ is a Hilbert space, then $||x||$ is equal to the root of the inner product $\langle x,x\rangle$, $||x_j||$ is equal to the root of $\langle x_j,x_j\rangle$, so the root of $\langle x_j,x_j\rangle$ converges to root of $\langle x,x\rangle$. Does this mean $\langle x_j,x_j\rangle$ converges to $\langle x,x\rangle$?
For my main question, I hope to get a counterexample if $\{x,x1,x2,\ldots\}$ is a subset of a Banach space. Thanks a lot.