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Math Help - Inner product space

  1. #1
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    Inner product space

    How ca I prove that in a real/complex vector space V
    ||x + y||^{2}+ ||x - y||^{2} = 2(||x||^{2} + ||y||^2)
    Also If I consider the parallelogram
    with adjacent sides OP; OQ where P is the point (x1; x2); Q is the point (y1; y2) and O is
    the origin.What does this say about a parallelogram in the plane?
    This is an exercisse from LINEAR ALGEBRA AND ITS APPLICATIONS David C. Lay.
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  2. #2
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    Quote Originally Posted by StefanM View Post
    How ca I prove that in a real/complex vector space V
    ||x + y||^{2}+ ||x - y||^{2} = 2(||x||^{2} + ||y||^2)
    Just consider this: \|x+y\|^2=(x+y)\cdot(x+y).
    Now expand to get x\cdot x+2x\cdot y+y\cdot y.
    Do that for \|x-y\|^2 and add.
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  3. #3
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    I was confuse about the fact that  ||x^{2}||=<x,x>...How can I solve the parallelogram problem?
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  4. #4
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    Quote Originally Posted by StefanM View Post
    I was confuse about the fact that  ||x^{2}||=<x,x>...How can I solve the parallelogram problem?
    You asked about proving \|x+y\|^2+\|x-y\|^2=\|x\|^2+2\|x\|\|y\|+\|y\|^2.
    That what I was addressing.
    It is a fact that \|x+y\|^2=<x+y,x+y>, which can be expanded.

    \|x+y\|~\&~\|x-y\| are the lengths of the diagonals of a parallelogram in the plane.
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