Let $x_n \to x$ weakly. My question is: does it hold that $\|x_n\|\to \|x\|$?
I haven't been able to work out the answer and I'd appreciate help with it but here are my thoughts:
Given the inverse $\Delta$-inequality: $|\|x_n\|-\|x\|| \le \|x_n -x\|$ it's clear that if they converge strongly then $\|x_n\|\to \|x\|$.
If the norm $\|.\|$ was continuous then it would hold but I suspect the norm is not continuous in the weak topology (although unfortunately I cannot seem to argue why this is so. Any hints appreciated).
My goal therefore is to show that $\|x_n \| \not \to \|x\|$ by finding an example of a Banach space $X$ and a sequence $x_n$ with $x_n \to x$ weakly but not $\|x_n\|\to \|x\|$.