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 $x_n$?