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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.

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    Thanks, but it was asked about six months ago...2012-05-21

1 Answers 1

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I know nothing about this problem, but I can say this:

Let's assume that completeness implies the equation, as you say. Now suppose that $\{u_n\}$ is not complete, but still orthonormal. Extend $\{u_n\}$ to an orthonormal system $\{v_n\}$ and wlog assume that $v_1$ is not among the $u_n$.

Then $a=\int_0^1|\int_0^xv_1(t)dt|^2dx>0$ and the sum taken over the $u_n$ is smaller than $1/2$ by at least $a$.

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    I find the first sentence confusing: You know nothing about it, but you solved it! +1. I always like it when in proving a two-way implication you get to use one direction to prove the other. I'll just note a way to see that a>0. If $a=0$, then $\int_0^xv_1(t)dt=0$ for almost all $x$, which by differentiation implies $v_1(x)=0$ a.e.2012-01-04