Operator in $\mathbb R^2$

Question

I am a bit confused, can someone help me with the following?
Is there an operator $T$ in $\mathbb{R}^2$ such that:
$\parallel u \parallel +\parallel v\parallel = \parallel T(u+v)\parallel$ for every $u,v\in \mathbb{R}^2$

Answer 1

There is no such linear mapping: Take $u=-v\ne0$, then $\|u\|+\|v\|>0$, but $T(u+v)=0$ be linearity.