2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67483A counterpart of the famous Bessel's inequality for orthornormal families in real or complex inner product spaces is given. Applications for some Gruss type inequalities are also provided.Classical Analysis and ODEsFunctional Analysis26D15, 26D10, 46C05A Counterpart of Bessel's Inequality in Inner Product Spaces and Some Gruss Type Related Resultstext