norm of a linear operator and dual

[Cathy]

Hello, all.
I am doing a problem and have some trouble with it. Any idea is greatly 
appreciated.
T is a linear operator from normed linear space X to a norm linear space Y.
I need to prove 
||T|| = Sup{<Tx,y*>: ||x||<=1,||y*||<=1,x in X, y* in Y*}

It is easy to get ||T||>= Sup<Tx,y*>. But the other direction of the inequality 
troubles me.

Cathy