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.

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

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?

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?

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?

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

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.