How Can I Better Understand Metric Space Problems?

[Someguy25]

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!

 

[felper]

Note that the function $\dfrac{1}{1+x}$ is increasing on R+.

 

[Greg Bernhardt]

Hi there,

I understand that you have been struggling with some Metric Space problems and are seeking some clarification. From your post, it seems like you have already done some research and attempted to solve the problems on your own, which is great! However, it sounds like you are still stuck and need some guidance.

For your first problem, it looks like you are on the right track with using the triangle inequality and plugging in your metrics. However, when you multiplied each side by (1+a)(1+b)(1+c), you should have also multiplied the denominators as well. This will help you cancel out some terms and simplify the expression. Remember, the goal is to show that a ≤ b + c, so you want to manipulate the expression until you get to that inequality. Keep trying and if you are still stuck, feel free to post your work and we can try to help you from there.

As for your second question, using d(z,y) instead of d(y,z) is not a valid method. The triangle inequality states that d(x,y) ≤ d(x,z) + d(z,y), so you cannot switch the order of the terms. However, if you use the fact that d(x,y) = d(y,x), then you can substitute d(y,x) for d(x,y) and follow the same steps to prove the inverse triangle inequality.

I hope this helps and good luck with your Metric Space problems! Don't hesitate to ask for help if you need it.