Some norm stuff

October 1st, 2007 ~ Posted in: Mathematics

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 P is a positive definite symmetric matrix, then define the quadratic norm \|P\|_x = (x^tPx)^{\frac{1}{2}}. Show that any norm on \R^n is equivalent to some quadratic norm, with constants 1 and \sqrt{n}:

\|x\|_P \leq \|x\| \leq \sqrt{n}\|x\|_P
Mathematics ~

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