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Math Help - Sequence Cauchy in L2?

  1. #1
    Junior Member
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    Sequence Cauchy in L2?

    Is the sequence {sin(nx)} Cauchy in L2 over [0,pi]? Why or why not?

    Under the L2 norm, this would amount to the integral of |sin(nx)-sin(mx)|^2 from 0 to pi being bounded by small epsilon for all n,m > N, right?

    I can't figure this guy out. Thanks for any help in advance.
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  2. #2
    Super Member
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    \{ \sin (nx) \} is an orthogonal set in L^2([0, \pi ]) (prove it) and so remember that if u and v are orthogonal vectors \Vert u + v \Vert ^2 = \Vert u-v \Vert ^2 = \Vert u \Vert ^2 + \Vert v \Vert ^2 (Because L^2([0, \pi ]) is a Hilbert sapce).
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