Prove if H is a separable Hilbert space and $\displaystyle \{y_\alpha\}_{\alpha\epsilon A}$ is a set of pairwise orthogonal vectors in H such that for any $\displaystyle \alpha\epsilon A, 0<\|y_\alpha\|\leq b$ for some constant b, then $\displaystyle \sum_{\alpha\epsilon A}|<x,y_\alpha>|^2\leq b^2\|x\|^2$ for any $\displaystyle x \epsilon H.$

What I have:

$\displaystyle \sum_{\alpha\epsilon A}|<x,y_\alpha>|^2\leq\sum_{\alpha\epsilon A}(\|x\|\|y_\alpha\|)^2 $ by Schwarz Inequality

$\displaystyle =\sum_{\alpha\epsilon A}\|x\|^2\|y_\alpha\|^2$

$\displaystyle \leq\sum_{\alpha\epsilon A}\|x\|^2b^2$

$\displaystyle =\|x\|^2b^2$

I just want to make sure that I can drop the sum. I think I can do this since the $\displaystyle y_\alpha$ are pairwise orthogonal, but I am not sure.

Thank you for your help.