Norm on vector space

william_33

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Mar 4, 2013
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If f\displaystyle f is a linear map on RpRq\displaystyle \mathbb{R}^p \to \mathbb{R}^q, define fpq=sup{f(x)Rp,x1}.\displaystyle ||f||_{pq} = sup\{||f(x)||\in \mathbb{R}^p, ||x|| \le 1\}.

Show that the mapping ffpq\displaystyle f\to ||f||_{pq} defines a norm on the vector space δ(Rp,Rq)\displaystyle \delta (\mathbb{R}^p, \mathbb{R}^q) of all linear functions on RpRq.\displaystyle \mathbb{R}^p \to \mathbb{R}^q. Show that f(x)fpqx\displaystyle ||f(x)||\le ||f||_{pq}||x|| for all xRp.\displaystyle x\in\mathbb{R}^p.

I don't know how to prove this. Can anyone help me please?
 
For the first part, what condition of being a norm are you having issue with?

Second: Let ||x||=a. Then f(x) = f(a*y) = af(y) where ||y||=1. Then ||f(x)|| = ||af(y)|| = a||f(y)||. So, convince yourself that it is enough to show the inequality is true when ||x||=1.
 
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