Q:

A simple question of a property of norms

For a normed space $X$, when does $$ \lim_{n \to \infty} \| x_n \|_X  = \| x \|_X$$ imply $$ \lim_{n \to \infty} \| x_n - x \|_X = 0 \;\;?$$

A:

Following on Thomas Andrews' comment, suppose $X$ has the property that $\lim_n\|x_n\|=\|x\|$ implies that $\lim_n\|x_n-x\|=0$. By considering constant sequences, this shows that $\|x\|=\|y\|$ implies $x=y$. Then $X$ is one-dimensional: indeed fix $x_0\in X$ with $x_0\ne0$. Write $c=\|x_0\|$. For any other nonzero $x\in X$, $\|x\|=\|\,\|x\|/c\,x_0\|$, so $x=\|x\|/c\,x_0$. 
Now using Tao's example, we consider the sequence $\{(-1)^nx_0\}$. As $\|(-1)^nx_0\|=\|x_0\|$, we get $\lim_n\|(-1)^nx_0-x_0\|=0$. The only way this can converge is when $x_0=0$.
So the only possible space with this property is the trivial space $\{0\}$. 

