I need a little confirmation on this one, it seemed a bit too easy for me, which makes me think I assumed too much:
Let T be a linear operator on an inner product space V, suppose ||T(x)||=||x|| for all x in V. Prove T is injective.
Here is my proof:
let x,y exist in V such that T(x)=T(y).
So ||T(x)||=||x|| implies
||T(x)||=||T(y)|| implies
||x||=||y|| implies
x=y. (Its this implication Im suspicious of).
Any help please?