Let $(X,d)$ be a metric space such that the set $X$ is also a vector space over the field $\mathbb{R}$ of real numbers or the field $\mathbb{C}$ of complex numbers. Then the following holds: 

> 
> If the metirc $d$ is induced by a norm on $X$, then we must have 
>  $$d(u+x,u+y) = d(x,y) \ \mbox{ for all } \ u, x, y \in X,$$
>  and 
>  $$d(\alpha x, \alpha y ) = \vert \alpha \vert \cdot d(x,y) \ \mbox{ for all } \ x, y \in X \ \mbox{ and for all scalars } \ \alpha.$$
> 
> 
> 

Are the above conditions sufficient also for the metric $d$ to be induced by a norm?