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I am trying to prove some defined function of two points if it is a metric or not. From the properties, I am having hard time showing $$\delta (p,q) \le \delta (p,r)+\delta (r,q)$$ for any $p,q,r$ in a metric space $X$.

I for some examples such as $$\delta (p,q) := (x^2-y^2)$$ I was able to come with some examples that doesn't satisfy this, therefore this function is not a metric.

However, the following two examples are not easy for me to see if it is a metric or not. $$\delta _1(p,q) := \sqrt{|p-q|}$$ and $$\delta _2(p,q):= {|p-q| \over {1+|p-q|}}$$.

If the explanation in the following is answered I would be delighted.

1),Is there an algebraic way to show these without taking too much time ?

2),Is it ok if I only checked two cases where p

I am self teaching analysis, and I have not taken topology. I only know pretty much up abstract algebra, so please keep that in mind.

hyg17
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  • Could you show me the proof, or where I can find the proof of this? Is it a famous theorem? – hyg17 May 09 '13 at 07:58

1 Answers1

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For $\delta_1$, use that $(a+b)^2 \geq a^2 + b^2$ for non-negative $a,b$.

For $\delta_2$, just cross multiply and simplify the inequality you need to prove, keeping in mind that the denominators are strictly positive.

ronno
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  • Thanks. I am still wondering if there is a faster way to decide. Do we always have to algebraically confirm ? – hyg17 May 09 '13 at 07:59
  • If you're given a function defined by a formula, it's natural to expect that you'd have to use that formula to verify some property of the function. Sometimes you can notice tricks that help your calculations, for example, what @julien mentions in his comments. – ronno May 10 '13 at 05:58