Proving Axioms of Norms: A Step-by-Step Guide

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In summary, the conversation is about proving the axioms of norms and using a given definition of the norm to prove its properties. The participants also discuss how to show that a particular example satisfies the axioms, and how to prove the convergence of a sum using the definition of the norm. The conversation ends with a question about using this proof to show the convergence of a specific sum involving a square matrix A.
  • #1
juniorgirl06
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ok guys.. i would really appreciate any help you may have..

Anyone know how to prove the axioms of norms? there are 3 of them.
 
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  • #2
What properties of the norm exactly are you trying to prove? Many of the basic properties will follow directly from the definitions. Show what you've got so far and you'll have a better chance of getting help.
 
  • #3
You can't prove an axiom, by definition.

Perhaps you are asking how you can show that a particular example satisfies a set of axioms?
 
  • #4
See. that's what I thought too..that you can't prove it b/c it is true by definition., but my prof seems to want us to prove it with a given definition of the norm.

he defines it to be , for a n x m matrix A, the norm of A=(sqrt(nm))(max{abs.val(aij)}), where the ij are subscripts.

I guess he wants us to show how this particular definition of the norm is in fact a norm. Any suggestions?
 
  • #5
Yes, just do it! Can you see why norm(a+b) <= norm(a) + norm(b)? It's a "just do it proof", in the sense of Polya.
 
  • #6
k here's some more I am having trouble with..


Prove the following test for convergence (using the above definition of the norm): If the sum, from k=0 to infinity, of the norm of Ak (where k is the subscript) converges then the sum of Ak) where k is the subscript) converges.

And how do i use this to prove that the sum from k=0 to infinity of Ak/k! ( where k in the numerator is an exponent) always converges for any square matirs A?
 

1. Can you explain what an axiom of norms is?

An axiom of norms is a fundamental principle or statement that serves as a basis for a set of norms or rules. It is a starting point that is accepted without proof and is used to derive other norms or rules.

2. What is the importance of proving axioms of norms?

Proving axioms of norms is important because it provides a solid foundation for a set of norms or rules. It ensures that the norms are logically consistent and can be accepted as true. Proving axioms also allows for a better understanding of the underlying principles and can help identify any potential flaws or loopholes in the norms.

3. How do you go about proving axioms of norms?

The process of proving axioms of norms involves carefully examining the underlying principles and using logical reasoning and mathematical techniques to show that the norms derived from the axioms are valid. This may involve constructing counterexamples or using deductive reasoning to show that the axioms lead to consistent and meaningful results.

4. Can you provide an example of proving an axiom of norms?

One example of proving an axiom of norms is the axiom of transitivity in mathematics, which states that if x is greater than y and y is greater than z, then x is greater than z. This axiom can be proven by constructing a logical argument and using mathematical equations to show that the axiom holds true for all possible scenarios.

5. Are there any challenges in proving axioms of norms?

Yes, there can be challenges in proving axioms of norms, as it requires a deep understanding of the underlying principles and the ability to think critically and logically. It may also involve complex mathematical calculations and the consideration of different scenarios, which can be time-consuming and require careful attention to detail.

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