Consider an alternative definition for the norm of a vector given by
- Which properties of Theorems 12.4 and 12.5 (pages 453-454 of Apostol) are still valid with this new definition?
- Draw a figure which shows the set of points of norm 1 with this new definition.
- Claim. All of the properties of Theorems 12.4 and 12.5 hold with this new definition.
Proof. For Theorem 12.4(a), if then there is some so . Therefore .
For Theorem 12.4(b), if then for all . Therefore and so .
For Theorem 12.4(c), we compute
For Theorem 12.5 we have
- We have the following figure,