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Let $\{u_n\}$ be an orthonormal system in $L^2([0,1])$, prove that $\{u_n\}$ is complete iff

$$ \sum_{n=1}^\infty \intop_0^1 \left|\intop_0^x u_n(t)\;dt\right|^2 dx = 1/2.$$

It should be noted that in the previous clause I proved that $\{u_n\}$ is complete iff

$$\forall x\in [0,1]:x=\sum_{n=1}^\infty \left| \intop_0^x u_n(t)\;dt\right|^2$$

and the two are probably related.

Proving that completeness if $\{u_n\}$ implies the equation is a simple consequence of this. I'm stuck on the second direction though, any hints would be appreciated.

  • 0
    This question is discussed here: http://www.matematicamente.it/forum/completeness-of-orthonormal-systems-in-l-2-t59695.html#p4237632012-05-21
  • 0
    Thanks, but it was asked about six months ago...2012-05-21

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