Convergence under a Hilbert space Let $\{\varphi_n\}_{n=1}^\infty$ be an orthonormal sequence (not necessarily a basis) in a Hilbert space. Let  $\{\lambda_n\}_{n=1}^\infty$ be a sequence of numbers
Define $T:H\to H$ by $Tx=
\sum_{n=1}^{\infty} \lambda_n\langle x, \varphi_n\rangle \varphi_n$
Show that if $sup_n|λ_n| < \infty$ then $\sum_{n=1}^{\infty} \lambda_n\langle x, \varphi_n\rangle \varphi_n$ converges for all
$x \in H$.
Well I have no clue for where to begin. Does it have something to do with Gram–Schmidt process?

Let $Tx=\sum_{n=1}^\infty \lambda_n (x, \varphi_n)_H \varphi_n$, then $T_N x \rightarrow T$ in $\mathcal{H}$, where $T_Nx=\sum_{n=1}^N \lambda_n (x,\varphi_n)_H \varphi_n$. By Bessel inequality we have
$\displaystyle \sum_{n=1}^N |(x,\varphi_n)_H|^2 \leq ||x||^{2}_H$
it's is valid for any orthonormal system. Therefore
$\displaystyle ||(T-T_N)x||_{H}^2 = \sum_{n > N} |\lambda_n(x, \varphi_n)_H|^2 \leq \sup_{N\in \mathbb{N}} |\lambda_N|^2 \sum_{n >N} |(x,\varphi_n)_H|^2 \leq \sup_{N \in \mathbb{N}}|\lambda_N|^2$
