How Can You Prove the Axioms of Norms and Use Them to Test Convergence?

[juniorgirl06]

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.

 

[shmoe]

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.

 

[James R]

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?

 

[juniorgirl06]

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?

 

[matt grime]

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.

 

[juniorgirl06]

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?