# Show that ||x||= d(x,0) defines a norm on V

• Oct 21st 2011, 01:26 PM
wopashui
Show that ||x||= d(x,0) defines a norm on V
Let V be a vector space, and let d be a metric on V satisfying $\displaystyle d(x,y) = d(x-y,0)$and$\displaystyle d(ax,ay) = |a|d(x,y)$for every $\displaystyle x,y \in V$ and every scalar a. Show that $\displaystyle ||x||=d(x,0)$ defines a norm on V (that has d as it standard metric). Give an example of a metric on the vector space R that fails to be associated with a norm in this way.
• Oct 21st 2011, 04:23 PM
Drexel28
Re: Show that ||x||= d(x,0) defines a norm on V
Quote:

Originally Posted by wopashui
Let V be a vector space, and let d be a metric on V satisfying $\displaystyle d(x,y) = d(x-y,0)$and$\displaystyle d(ax,ay) = |a|d(x,y)$for every $\displaystyle x,y \in V$ and every scalar a. Show that $\displaystyle ||x||=d(x,0)$ defines a norm on V (that has d as it standard metric). Give an example of a metric on the vector space R that fails to be associated with a norm in this way.

What have you tried? Is there a particular place you are having trouble?
• Oct 21st 2011, 06:24 PM
Deveno
Re: Show that ||x||= d(x,0) defines a norm on V
for a counter-example, use the discrete metric. which hypothesis does this fail to satisfy, and why does it fail to yield a norm?
• Oct 21st 2011, 08:57 PM
wopashui
Re: Show that ||x||= d(x,0) defines a norm on V
Quote:

Originally Posted by Drexel28
What have you tried? Is there a particular place you are having trouble?

I have trouble starting the proof, do I just use the definition of norm to show that d(x,0)=|x-0| satisfy the properites of norm?
• Oct 21st 2011, 09:09 PM
Drexel28
Re: Show that ||x||= d(x,0) defines a norm on V
Quote:

Originally Posted by wopashui
I have trouble starting the proof, do I just use the definition of norm to show that d(x,0)=|x-0| satisfy the properites of norm?

It seems like you are thinking this norm is given. You aren't DEFINING $\displaystyle d$ you are DEFINING $\displaystyle \|\cdot\|$, right? So, you need to show that the norm $\displaystyle \|\cdot\|$ defined by $\displaystyle \|x\|=d(x,0)$ is really a norm. For example $\displaystyle \|ax\|=d(ax,0)=d(ax,a0)=|a|d(x,0)=|a|\|x\|$, etc.
• Oct 21st 2011, 09:54 PM
Deveno
Re: Show that ||x||= d(x,0) defines a norm on V
ok, so you've proved d is positive scalable, but is it subadditive? does it separate points? i don't want any of those cheap semi-norms.