the norm for a linear transformation T on R^n is ||T|| = {max|T(x)| : |x|<= 1}
i need to show this is equivalent to ||T|| = {max|T(x)| : |x| = 1}
Let $\displaystyle B= \{ x\in \mathbb{R} ^n : \| x \| \leq 1 \}$ and $\displaystyle S=\{ x\in \mathbb{R} ^n : \| x \| =1 \}$ then $\displaystyle \sup _{B} \{ \| T(x) \| \} = \sup _{B} \{ \| x \| \| T\left( \frac{x}{ \|x\| } \right) \| \} \leq \sup_{S} \{ \| T(y) \| \} $
where the last inequality follows since $\displaystyle x\in B$ and $\displaystyle \frac{x}{ \| x\| } \in S $. The other inequality is immediate since $\displaystyle S \subset B$.
PS. Notice that this proof only requires a normed vector space.