show that a normed vector space $(V,\Vert . \Vert)$ can become a metric space

Question

with a distance defined by:

$d(x,y) = \Vert (x-y)\Vert$

I know that

Every normed space is a metric space, but not the other way round

and:

Metric spaces are much more general than normed spaces

and:

the principles that distinguish a norm from a metric are 1) translation invariance and 2) homogeneity.

At this point what I have to write formally?

thanks

Answer 1

It's just a matter of noticing that:

• $d(x,y)=0\iff\|x-y\|=0\iff x=y$.
• $d(y,x)=\|y-x\|=\|(-1)\times(y-x)\|=|-1|.\|x-y\|=\|x-y\|$;
• $d(x,z)=\|x-z\|=\|x-y+y-z\|\leqslant\|x-y\|+\|y-z\|=d(x,y)+d(y,z)$.