Norm on vector space

[william_33]

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

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

I don't know how to prove this. Can anyone help me please?

 

[daon2]

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.