Q:

$\sum_{n=1}^\infty \langle x,e_n\rangle\langle y,e_n\rangle$

$H$ Hilbert space. $\{e_n\}$ orthonormal basis and $x,y\in H$.
In a proof I have $\sum_{n=1}^\infty |\langle x,e_n\rangle|^2\leq 1$ and $\sum_{n=1}^\infty |\langle y,e_n\rangle|^2\leq 1$.
Is it true that $\sum_{n=1}^\infty \langle x,e_n\rangle\langle y,e_n\rangle \leq 1$?

A:

I see no reason why $\sum_{n=1}^{\infty}\langle x,e_n\rangle\langle y,e_n\rangle$ should be a real number, so it may not be meaningful to say that it is $\leq 1$. 
But in any case,
$$ \Big|\sum_{n=1}^{\infty}\langle x,e_n\rangle\langle y,e_n\rangle\Big|\leq \Big[\sum_{n=1}^{\infty}|\langle x,e_n\rangle|^2\Big]^{\frac{1}{2}}\Big[\sum_{n=1}^{\infty}|\langle y,e_n\rangle|^2\Big]^{\frac{1}{2}}\leq 1$$
by the Cauchy-Schwarz inequality in $\ell^2(\mathbb{N})$.

