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Metric Space Explanation

  1. Oct 16, 2011 #1
    Hey All,

    I have been working on some Metric Space problems for roughly 20hrs now and I cannot seem to grasp some of these concepts so I was hoping someone here could clear a few things up for me. My first problem is detailed below...

    I have the following metric...

    d(x,y) = d(x,y)/(1 + d(x,y)

    Now I did a search on google and found a few examples on how to solve this and even one on these forums; however, nothing seems to really make sense to me. On this post https://www.physicsforums.com/showthread.php?t=527353", I do not understand how to finally cancel terms out. I did the following..

    a = d(x,y)
    b = d(x,z)
    c = d(y,z)

    Plugging into the triangle inequality we get...

    a ≤ b + c

    From here we can use our metrics and get...

    a/(1+a)≤ b/(1+b) + c/(1+c)

    Now in the post it talks about multiplying each side by (1+a)(1+b)(1+c) If I do this I arrive at...

    a(1+a)(1+b)(1+c)/((1+a)(1+a)(1+b)(1+c)) ≤ b(1+a)(1+b)(1+c)/((1+b)(1+a)(1+b)(1+c) + c(1+a)(1+b)(1+c)/((1+c)(1+a)(1+b)(1+c)

    From here it just becomes a mess to me and I feel like I am not making any progress. Can someone point out to me what I am doing wrong and where I should be going with this please?

    My next question is simple...

    For proving the inverse triangle inequality I used d(z,y) → d(y,z) and substituted used that to switch out each value of x and y so for instance...

    d(x,y) ≤ d(x,z) + d(y,z) → d(x,z) ≤ d(x,y) + d(z,y)

    I was wondering if this was a valid method?

    Thank you all in advance for you help!!
     
    Last edited by a moderator: Apr 26, 2017
  2. jcsd
  3. Oct 17, 2011 #2
    Note that the function [itex]\dfrac{1}{1+x}[/itex] is increasing on R+.
     
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