Some norm stuff
Note that the unit ball of a norm satisfies the following properties: it is symmetric about the origin, convex, closed, bounded, and has a nonempty interior. Now show that any set with these properties is the unit ball of some norm (come up with an exact formula).
If 
is a positive definite symmetric matrix, then define the quadratic norm 
. Show that any norm on 
is equivalent to some quadratic norm, with constants 1 and 
:
Possibly relevant posts:
- Geodesics and Metrics (4/22/2005)
- Inner products (6/7/2006)
- Some Schatten norm stuff (6/17/2008)