Sequence Cauchy in L2?

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• July 15th 2009, 01:03 PM
joeyjoejoe
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.
• July 15th 2009, 01:21 PM
Jose27
$\{ \sin (nx) \}$ 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).