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Math Help - Parseval's Identity

  1. #1
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    Parseval's Identity

    I'm trying to prove Parseval's Identity.

    "Let \{v_1, ..., v_n\} be an orthonormal basis for V. For any x, y \in V prove that \langle x,y \rangle = \displaystyle\sum^n_{i=1} \langle x,v_i \rangle \overline{\langle y,v_i \rangle}."

    I'm really not sure how to go about this. I've played around with the summand, including taking x as a linear combination of the v_i and breaking it down, but that actually just gets us right back to \langle x,v_i \rangle.

    I'd like some hints, but only hints. Thanks.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Parseval's Identity

    Quote Originally Posted by AlexP View Post
    I'm trying to prove Parseval's Identity.

    "Let \{v_1, ..., v_n\} be an orthonormal basis for V. For any x, y \in V prove that \langle x,y \rangle = \displaystyle\sum^n_{i=1} \langle x,v_i \rangle \overline{\langle y,v_i \rangle}."

    I'm really not sure how to go about this. I've played around with the summand, including taking x as a linear combination of the v_i and breaking it down, but that actually just gets us right back to \langle x,v_i \rangle.

    I'd like some hints, but only hints. Thanks.
    Just do what's natural. By definition you have that x=\langle x,v_1\rangle v_1+\cdots+\langle x,v_n\rangle v_n and y=\langle y,v_1\rangle y_1+\cdots+\langle y,v_n\rangle v_n and so



    \displaystyle \langle x,y\rangle=\left\langle \sum_{i=1}^{n}\langle x,v_i\rangle v_i,\sum_{j=1}^{n}\langle y,v_i\rangle v_i\right\rangle


    so what if we expand by sequilinearity?
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  3. #3
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    Re: Parseval's Identity

    ok, I got it. I was trying to get the RHS into a useful form, rather than the LHS...not entirely sure why. But now in the future I'll remember to try both sides. Lesson learned. Thanks.
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