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Math Help - Inner product space law

  1. #1
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    Inner product space law

    In an inner product space, show that ||x+y|| ||x-y|| \leq ||x||^2 + ||y||^2

    Proof.

    I want to show that 2||x||^2 + 2||y||^2 \geq 2||x+y||||x-y||

    By the Parallelogram law, I know that 2||x||^2 +2||u||^2 = ||x+y||^2+||x-y||^2 \geq ||x+y||^2-||x-y||^2 =( ||x+y|| + ||x-y||)(||x+y|| - ||x-y||)

    But I don't seem to be able to get to it, am I starting this wrong? Thanks.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    In an inner product space, show that ||x+y|| ||x-y|| \leq ||x||^2 + ||y||^2
    Try squaring both sides, then expand \|x\pm y\|^2 as \langle x\pm y,x\pm y\rangle.
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