2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/221750We use Salem's method to prove that there is a lower bound for partial sums of series of bi-orthogonal vectors in a Hilbert space, or the dual vectors. This is applied to some lower bounds on $L^{1}$ norms for orthogonal expansions. There is also an application concerning linear algebra.Classical Analysis and ODEs42C15; 46C05text