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Math Help - Bessel's Inequality

  1. #1
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    Bessel's Inequality

    I'm having trouble with this problem. I've tried rewriting things in about every way I know how to but I haven't arrived at an answer. I'd appreciate some help.

    "Let V be an inner product space and let S=\{v_1, ..., v_n\} be an orthonormal subset of V. Prove that for any x \in V we have ||x||^2 \ge \displaystyle\sum^n_{i=1} |\langle x,v_i \rangle |^2."

    As a hint the book says to use the fact that x \in V can be written uniquely as w+w' where w \in W=\mbox{span}(S) and w' \in W^\perp, the orthogonal complement of W, and use the fact that for x, y orthogonal, ||x+y||^2 = ||x||^2 + ||y||^2.

    Thanks for any help.
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  2. #2
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    Re: Bessel's Inequality

    Quote Originally Posted by AlexP View Post
    I'm having trouble with this problem. I've tried rewriting things in about every way I know how to but I haven't arrived at an answer. I'd appreciate some help.

    "Let V be an inner product space and let S=\{v_1, ..., v_n\} be an orthonormal subset of V. Prove that for any x \in V we have ||x||^2 \ge \displaystyle\sum^n_{i=1} |\langle x,v_i \rangle |^2."

    As a hint the book says to use the fact that x \in V can be written uniquely as w+w' where w \in W=\mbox{span}(S) and w' \in W^\perp, the orthogonal complement of W, and use the fact that for x, y orthogonal, ||x+y||^2 = ||x||^2 + ||y||^2.

    Thanks for any help.
    As the hint gives

    \mathbf{x}=\mathbf{x}_{v}+\mathbf{x}_{v\perp}

    Now

    ||\mathbf{x}||^2=<\mathbf{x},\mathbf{x}>=<\mathbf{  x}_{v}+\mathbf{x}_{v\perp},\mathbf{x}_{v}+\mathbf{  x}_{v\perp}>=||\mathbf{x}_v||^2+||\mathbf{x}_{v \perp}||^2

    Remember that

    \mathbf{x}_{v}=\sum_{i=1}^{n}<\mathbf{x},\mathbf{v  }_i>\mathbf{v}_i \implies ||\mathbf{x}_v||^2=\sum_{i=1}^{n}|<\mathbf{x}, \mathbf{v}_i >|^2

    Just put these two facts together and remember that the modulus of a vector is always positive to finish.
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  3. #3
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    Re: Bessel's Inequality

    Wow. I was so close to completing it that it's painful that I didn't see it. Thanks.
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