Results 1 to 3 of 3

Math Help - Help with Metric Spaces?

  1. #1
    Junior Member
    Joined
    Oct 2009
    Posts
    25

    Post 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!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Assuming d_m is a metric d_{mb}:=d is also a metric since d(x,y)=0 iff d_m(x,y)=0 iff x=y and clearly d \geq 0. The commutativity follows immediately and so we only have to prove the triangle inequality which is equivalent to: 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 d_m? Is it the usual metric?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943
    Quote Originally Posted by Majialin View Post
    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 \mathbb{R}^n, because \forall~x,y\in\mathbb{R}^n, d_{mb}(x,y)<1.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Any metric spaces can be viewed as a subset of normed spaces
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: December 15th 2011, 04:00 PM
  2. [SOLVED] metric spaces
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: October 30th 2010, 11:55 AM
  3. metric spaces
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 20th 2010, 12:11 PM
  4. Metric Spaces ...... help!
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: December 16th 2009, 11:16 AM
  5. metric spaces
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 3rd 2009, 01:33 AM

Search Tags


/mathhelpforum @mathhelpforum