Consider an alternative definition for the norm of a vector given by
- Prove that this definition satisfies all of the properties of Theorems 12.4 and 12.5 (pages 453-454 of Apostol).
- Consider this definition in and draw the set of all points which have norm 1.
- If we defined
which of the properties of Theorems 12.4 and 12.5 would hold?
- Proof. For Theorem 12.4(a) we have
if since all of the terms in the sum are greater than or equal to 0, with at least one non-zero since .
For Theorem 12.4(b) if then .
For Theorem 12.4(c) we have
For Theorem 12.5 we have
- We have the following diagram
- Property 12.4(a) fails since if we take then , but .
Property 12.4(b) holds since if then .
Property 12.4(c) holds sinceProperty 12.5 holds since