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Let $ H $ be a Hilbert space, and let $\{x_k \}_{k\in \mathbb{N}}$ be an orthogonal subset of $H$. If for every $y\in H$, $\sum \left<x_k, y\right>$ converges, then $ \sum x_k $ converges too.

I found it as a theorem in functional analysis by Rudin (second edition, section 12.6 pag. 309) but I can't use the Banach-Steinhaus theorem because it is an advanced theorem (I have not seen it yet). Do you know a simpler proof of this theorem?

Sebastian Alvarez
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  • I'm voting to close this question as off-topic because it is some sort of take-home exam, see deleted post http://mathoverflow.net/questions/223985/convergent-series-of-orthogonal-vectors-in-hilbert-space – Will Jagy Nov 19 '15 at 16:57
  • @WillJagy It's a standard fact in functional analysis, shutting down the discussion of it just because someone on the planet put it on an exam is an overreaction. –  Nov 20 '15 at 06:03

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Let us define the partial sum $$z_n = \sum_{k = 1}^n x_k.$$ Then, we have for $m > n$ (by orthogonality) $$\|z_m - z_n\|^2 = \left\|\sum_{k = n+1}^m x_k\right\|^2 = \sum_{k = n+1} \|x_k\|^2.$$ Hence, $\{z_n\}$ is a Cauchy sequence in $H$ iff $\{\|x_k\|^2\}$ is summable. Hence, it remains to show that $\{\|x_k\|^2\}$ is summable.

Suppose that $\{\|x_k\|^2\}$ is not summable. Then, there is a non-negative sequence $a \in \ell^2$ with $$\sum_{k=1}^\infty a_k \, \|x_k\| = +\infty.$$ Now, we set $$y = \sum_{k=1}^\infty a_k \, \frac{x_k}{\|x_k\|}.$$ (This series converges by $a \in \ell^2$ and orthonormality of $\{x_k/\|x_k\|\}$.)

But, $$(y, z_n) = \sum_{k=1}^n a_k \, \|x_k\|$$ diverges in contradiction with your assumption.

Community
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gerw
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