2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67523Some new counterparts of Bessel's inequality for orthornormal families in real or complex inner product spaces are pointed out. Applications for some Gruss type inequalities are also empahsized.Classical Analysis and ODEsFunctional Analysis26D15, 26D10, 46C05Some New Results Related to Bessel and Gruss Inequalities for Orthogonal Families in Inner Product Spacestext