# Help with Metric Spaces?

• Oct 22nd 2009, 04:55 PM
Majialin
Help with Metric Spaces?
I have a two part question that I'm not quite sure how to approach... it goes as follows:

1. Prove that (R^n, d) is a metric space, where:
d_mb(x,y) =d_m(x,y)/(d_m(x,y) +1)

2. Find the unit ball B_M(0; 1) for M = (R^3, d), where d is the distance function defined in 1.

Any help would be appreciated! Thank you!
• Oct 22nd 2009, 05:14 PM
Jose27
Assuming $\displaystyle d_m$ is a metric $\displaystyle d_{mb}:=d$ is also a metric since $\displaystyle d(x,y)=0$ iff $\displaystyle d_m(x,y)=0$ iff $\displaystyle x=y$ and clearly $\displaystyle d \geq 0$. The commutativity follows immediately and so we only have to prove the triangle inequality which is equivalent to: $\displaystyle d_m(x,z)(d_m(x,y)+1)(d_m(y,z)+1) \leq (d_m(x,z)+1)(d_m(x,y)+1)$ which is clearly true.

For the second one, which is $\displaystyle d_m$? Is it the usual metric?
• Oct 22nd 2009, 06:24 PM
redsoxfan325
Quote:

Originally Posted by Majialin
I have a two part question that I'm not quite sure how to approach... it goes as follows:

1. Prove that (R^n, d) is a metric space, where:
d_mb(x,y) =d_m(x,y)/(d_m(x,y) +1)

2. Find the unit ball B_M(0; 1) for M = (R^3, d), where d is the distance function defined in 1.

Any help would be appreciated! Thank you!

2. It is all of $\displaystyle \mathbb{R}^n$, because $\displaystyle \forall~x,y\in\mathbb{R}^n$, $\displaystyle d_{mb}(x,y)<1$.