In Hilbert space why does convergence in weak topology $x_n$ to $x$ imply that $x_n$ converges to $x$ in norm?
Thank you very much for your answers. What if I put a condition on weak convergence i.e., suppose it also holds $\lim_{n\to \infty } \lVert x_n\rVert \to \lVert x\rVert$ then can I say that $x_n$ converges in norm?
I doubt if it always holds!
I think to understand this I need bit more explanation because I am struggling with the understanding of weak topologies. Thank you