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Thread: Convergent Series and Orthonormal Sets

  1. #1
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    Convergent Series and Orthonormal Sets

    What is an example of a convergent series $\displaystyle \sum <x, e_{k}>e_{k}$ that does not have a sum of x?
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  2. #2
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    Re: Convergent Series and Orthonormal Sets

    If you just chose thing pretty much at random, you would probably get such an example. It is only as long as the vectors $\displaystyle e_k$ form an orthonormal basis for whatever vector space you are talking about that this sum will be equal to x.

    An example is $\displaystyle e_1= <2, 0>$, $\displaystyle e_2= <0, 3>$, and $\displaystyle x= <1, 1>$.
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