Thread: 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 and for every and every scalar a. Show that 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.

Originally Posted by wopashui

Let V be a vector space, and let d be a metric on V satisfying and for every and every scalar a. Show that 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?

for a counter-example, use the discrete metric. which hypothesis does this fail to satisfy, and why does it fail to yield a norm?

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?

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 you are DEFINING , right? So, you need to show that the norm defined by is really a norm. For example , etc.

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.