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Math Help - Show that in an inner product space

  1. #1
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    Show that in an inner product space

    Dear Colleagues,

    Could you please help me in solving the following problem:
    Show that in an inner product space, x\bot y if and only if ||x+\alpha y||\geq ||x|| for all scalars \alpha.

    Remark: I have already proved that x\bot y implies that ||x+\alpha y||\geq ||x|| for all scalars \alpha, it remains the converse.


    Regards,

    Raed.
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  2. #2
    Super Member girdav's Avatar
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    Take the squares and expand the inner product.
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  3. #3
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    Thank you very much for your reply, I have already done that and the result is
    2Re(\alpha {\bar} \langle x,y \rangle) + |\alpha|^{2}\langle y,y \rangle \geq 0, where Re denotes the real part of a complex number.

    But what then?
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  4. #4
    Super Member girdav's Avatar
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    You can divide by |\alpha|^2.
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