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Math Help - Inner Product Space

  1. #1
    Super Member Deadstar's Avatar
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    Inner Product Space

    Let V be a real inner product space. Show that

    (x , y) = \frac{1}{4}(||x + y||^2 - ||x - y||^2)

    for all x , y \in V.

    Not really sure where to start with this...
    Do i start by proving that (x , y) = (y , x)? which i can see because of the absolute value signs (is that what the || lines are called..?) that it is.

    Then is it just showing that \lambda x + \mu y , z = \lambda (x,z) + \mu (y,z)
    and also
    (x , x) = 0
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  2. #2
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    \left\| {x + y} \right\|^2 =  \left( {x + y,x + y} \right) = (x,x) + 2(x,y) + (y,y) = \left\| x \right\|^2  + 2(x,y) + \left\| y \right\|^2
    \left\| {x - y} \right\|^2 = \left( {x - y,x - y} \right) = (x,x) - 2(x,y) + (y,y) = \left\| x \right\|^2  - 2(x,y) + \left\| y \right\|^2
    Can you finish?
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  3. #3
    Super Member Deadstar's Avatar
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    Oh yeah i see now.
    Yeah can finish, ||x||'s and ||y||'s cancel leaving 4(x , y) so need a 1/4 to make it = (x , y). Thanks. Think i was just confused with what the question was asking. Been doing a lot of inner product proofs involving what i said i thought had to be done in my first post.
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