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Math Help - problem involving norm

  1. #1
    Member
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    problem involving norm

    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?
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by dannyboycurtis View Post
    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.
    If x=-y then \|x\| = \|y\| so \|x\|=\|y\| does not imply x=y.

    As T is a linear operator, T(x)=T(y) implies T(x-y)=T(x)-T(y)=0. Hence we have \|T(x-y)\| = 0. I am sure you can take it from here.
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