Metrics from Operator Norms

Question

Let $X$ be a Hilbert space and $(\cdot,\cdot)_X$ be the inner product on $X$. It is well known that $|x|_X = \sqrt{(x,x)_X}$ is a norm on $X$ and $|x-y|_X$ is a metric on $X$. The norm on $X$ induces norms on $B(X)$, e.g. $||T|| = \sup_{x\in X}\{|Tx|_X:|x|_X=1\}$. Under what induced norms is $||T-V||$ a metric on $B(X)$?