P-adic metric Strong triangle inequality

In summary, the speaker is asking for a good resource on the P-adic metric strong triangle inequality and is wondering if a proof for the metric space using the strong triangle inequality can also use the normal inequality. The answer is that the strong triangle inequality implies the usual triangle inequality, so proving either one is sufficient.
  • #1
beetle2
111
0
Hi Guy's
I was wondering If anyone new of a good link about the P-adic metric Strong triangle inequality. I am trying to constuct a proof to show that the p-adic is a metric space.
Must the proof use the Strong triangle inequality ie

[tex] d(a,c)\leq [/tex]max[tex]{d(a,b),d(b,c)}[/tex]

or can it use the normal inequality ?



[tex] d(a,c)\leq d(a,b),d(b,c)[/tex]
[tex] d(a,c)=\mida-c\mid=\mid(a-b)-(c-b)\mid[/tex]

regards
 
Last edited:
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  • #2
Of course the strong triangle inequality implies the (usual) triangle inequality. So if you prove either one of them, you are done.
 

1. What is the P-adic metric Strong triangle inequality?

The P-adic metric Strong triangle inequality is a mathematical concept that states that in a P-adic metric space, the distance between three points is always less than or equal to the sum of the distances between any two of the points. In other words, it is a way to measure the distance between points in a space using a P-adic metric, which is a type of metric that is often used in number theory.

2. How is the P-adic metric Strong triangle inequality different from the regular triangle inequality?

The regular triangle inequality states that the distance between two points in a space is always less than or equal to the sum of the distances between any two other points in the space. However, the P-adic metric Strong triangle inequality is more strict, as it requires the distance between three points to be less than or equal to the sum of the distances between any two of the points. This is because the P-adic metric takes into account the unique properties of P-adic numbers, which are different from regular real numbers.

3. What are some applications of the P-adic metric Strong triangle inequality?

The P-adic metric Strong triangle inequality has many applications in number theory, algebraic geometry, and cryptography. It is used to study and analyze properties of P-adic numbers, which have important applications in number theory and algebraic geometry. It is also used in cryptography, as P-adic numbers can be used to create secure encryption algorithms.

4. How is the P-adic metric Strong triangle inequality proven?

The P-adic metric Strong triangle inequality can be proven using mathematical induction and the properties of P-adic numbers. First, it is proven for a base case of three points. Then, using the inductive hypothesis, it can be proven for any number of points. This proof relies on the fact that P-adic numbers have a unique property called ultrametricity, which is what allows the P-adic metric Strong triangle inequality to hold.

5. Are there any alternative forms of the P-adic metric Strong triangle inequality?

Yes, there are several alternative forms of the P-adic metric Strong triangle inequality, such as the reverse triangle inequality and the weak triangle inequality. These alternative forms have slightly different conditions or restrictions, but they all stem from the fundamental principle of the P-adic metric Strong triangle inequality that the distance between three points is always less than or equal to the sum of the distances between any two of the points.

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