Q:

Generalization of Bessel inequality

Let $H$ be a Hilbert space and suppose we have a sequence $\{x_n\}\subset H$ s.th. for every $h\in H$ we have $ \sum \left|\langle h, x_n\rangle\right|^2<\infty$. How to prove that there exists nonnegative $c$ s.th. for every $h\in H$
$\sum \left|\langle h, x_n\rangle\right|^2\le c\|h\|^2$ ?
I tried to use Bessel inequality, reformulate proof of Bessel inequality, Cauchy inequality, but it didn't work. Any hint ?

A:

By your assumption, the operator
$$
\Phi : H \to \ell^2 (\Bbb{N}), h \mapsto (\langle h, x_n\rangle)_{n \in \Bbb{N}}.
$$
By your assumption, this map is well-defined. It is easy to verify that it has closed graph. Now, the closed graph theorem yields your claim.

