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Let $\{x_n\}$ be a sequence of pairwise orthogonal vectors in a Hilbert space $H$. Prove that the following are equivalent:

a) $\displaystyle\sum_{n=1}^\infty \|x_n\|^2<\infty$

b) $\displaystyle\sum_{n=1}^\infty x_n$ converges in the norm topology of $H$

c) $\displaystyle\sum_{n=1}^\infty (y,x_n)$ converges for each $y\in H$.

I got a implies b. On b implies c, Let $y\in H$. Then $(y,\sum_{n=1}^m x_n)=\sum_{n=1}^m (y,x_n)$ then take the limit as $m$ approaches infinity. Is this all that is needed? For c implies a, do we just replace $y$ with $x_n$?

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1 Answer 1

up vote 1 down vote accepted

Hint

For each $m\in\mathbb{N}$ consider $y=\sum\limits_{n=1}^m x_n$ and use that vectors $\{x_n:n\in\mathbb{N}\}$ are pairwise orthogonal. Think what will happen if $m\to\infty$.

P.S.

You can't replace $y$ by $x_n$, since $x_n$ can be diferent elements of $H$. Element $y$ can not be equal to all $x_n$ simultaneously!

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thank you. I figures it would not be that easy –  john May 30 '12 at 20:19
    
actually, is such a y in H. In assuming c, we do not know the series converges –  john May 30 '12 at 20:27
    
@john, good observation, see edits to my answer –  Norbert May 30 '12 at 20:48

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