Folks,

I have

Axiom 1:

Axiom 2: iff

Axiom 3: for

Axiom 4: The triangle inequality.

How do I get started on any of these axioms?

THanks

Printable View

- Nov 10th 2011, 10:27 AMbugatti79Prove that the modulus is a norm on C
Folks,

I have

Axiom 1:

Axiom 2: iff

Axiom 3: for

Axiom 4: The triangle inequality.

How do I get started on any of these axioms?

THanks - Nov 10th 2011, 10:33 AMDevenoRe: Prove that the modulus is a norm on C
is |z| ≥ 0, for any complex number z? certainly for any real a,b (why?). thus the positive square root is defined for such a number. why is this non-negative? these are just basic properties of real numbers.

when can |z| be 0? draw a picture. formalize your picture with a proof.

for axiom 3, write out α and z as α = c+id, z = a+ib. do the multiplication. take the norm of the product, and compare with the product of the norms. what do you find?

you have to use your definitions to prove axioms. - Nov 10th 2011, 11:12 AMbugatti79Re: Prove that the modulus is a norm on C
- Nov 10th 2011, 12:06 PMDevenoRe: Prove that the modulus is a norm on C
um....isn't the POSITIVE square root of a non-negative number always 0 or positive?

Quote:

|z| can be 0 only when the function z is 0, ie is the 0 function...?

z is not a "function". it's just a plain ol' complex number.

Quote:

I get

and

.......?

- Nov 10th 2011, 12:42 PMPlatoRe: Prove that the modulus is a norm on C
In order to prove the triangle inequality property of the complex norm you must prove some lemmas.

.

.

Once you have shown those are true then consider:

. - Nov 17th 2011, 11:22 AMbugatti79Re: Prove that the modulus is a norm on C