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Quick proof regarding inner products

  1. Jul 29, 2009 #1
    1. The problem statement, all variables and given/known data

    Let V be an inner product space. If <x,y> = <x,z> for all x in V, then prove that y = x.


    2. Relevant equations



    3. The attempt at a solution

    Since V is a vector space, it follows that (y - z) is an element of V. Since <x,y> = <x,z> for ALL x, we put x = (y-z). Then we have <y - z, y> - <y - z, z> = 0. Since inner products are conjugate linear in the second component, we can write:

    <y - z, y> - <y - z, z> = <y -z, y - z> = 0. I know that in general <w,w> = 0 if and only if w = 0. Thus we must have y - z = 0, implying that y = z.

    Is this correct?
     
  2. jcsd
  3. Jul 29, 2009 #2
    I think that you lose generality if you put x=(y-z).
    Probably the proof applies then to that particular case.
    Anyway, there is my proof:

    1. y = z + w
    2. (x, z + w) = (x, z)
    3. (x, z) + (x, w) = (x, z)
    4. (x, w) = 0

    now, since x can be any vector, the difference between y and z,
    namely w, must be a null vector and so y and z are equal.

    P.S. In the question you are asked to prove x=y
    but then you prove z=y so I guess there is a typo.

    ---
     
    Last edited by a moderator: Aug 6, 2009
  4. Jul 29, 2009 #3

    Dick

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    Your proof is just fine.
     
  5. Jul 29, 2009 #4
    Thanks Dick.
     
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