Given $u, v\in \mathbb R^d$, I am trying to figure out the factor $M(u,v)$ in the bound
$$\|u\otimes u - v\otimes v\| \leq M(u,v) \cdot \| u - v\|_2,$$
if that's possible. The first norm is the standard operator norm. I think it must be something like $M(u,v) = \|u\| + \|v\|$ but I'm having trouble not getting bogged down in the battle with the indices. I suppose that this is a fairly standard result but I couldn't find it so far.
I'm also interested in the generalization the Hilbert space case with
$(u \otimes v)(w) := u \cdot \langle v, w\rangle.$
The way I see it $$\|u\otimes u - v\otimes v\|=\sup_{||x||=1}\|\langle x,u\rangle u-\langle x,v \rangle v \|=\sup_{||x||=1}\|\langle x,u\rangle u-\langle x,u\rangle v+\langle x,u\rangle v-\langle x,v \rangle v \| \le \sup_{||x||=1}\|\langle x,u\rangle u-\langle x,u\rangle v\|+\sup_{||x||=1}\|\langle x,u\rangle v-\langle x,v \rangle v \| \le \|u-v\|\|u\|+\|u-v\|\|v\|$$ $$=(\|u\|+\|v\|)\|u-v\|$$