Linear transformations between normed linear spaces

[Ant farm]

Hi, ok I'm working with linear transformations between normed linear spaces (nls)

if T :X -> Y nls's is a linear transformation, we define the norm of T, ||T||: sup{||T|| : ||x||<=1}

I want to show that for X not = {0}
||T||: sup{||T|| : ||x|| = 1} frustratingly the books all assume that this step is obvious... I don't see how.

Intuitively I can see that is true, using the fact that (I think) T is a bounded function, and we can manipulate things to make the whole function be contained within a closed unit sphere...
Have always had a mental block with inf's and sups...don't know why...

 

[mathman]

Work with x/||x||. It will be obvious.