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?

It seems you mean that $T$ is well-defined. Let $x\in H$,
$$\|\sum_m^n \lambda_i\langle x,\phi_i\rangle\phi_i\|^2=\sum_m^n |\lambda_i\langle x,\phi_i\rangle|^2\leq(\sup_{n\geq1} \lambda_n)\sum_m^n |\langle x,\phi_i\rangle|^2\to0$$ 
