Can someone give me an idea, why $\forall x: \left<\sum_j \lambda_j \left< x,e_j\right> e_j,x\right>\geq 0$, where the $\lambda_j$'s are fixed, implies that all $\lambda_j$ are $\geq0$,?
(The $x$'s belong to a Hilbert space,the $e_j$'s are an orthonormal basis and the $\lambda_j$'s are real or complex .)