Improving the Cauchy–Schwarz Inequality

We highlight overlap as one of the simplest inequalities in linear space that yields a number of useful results. One obtains the Cauchy–Schwarz inequality as a special case. More importantly, a variant of it is seen to work desirably in certain singular situations where the celebrated inequality appears to be useless. The basic tenet generates a few other interesting relations, including the improvements over certain common uncertainty bounds. Role of projection operators in modifying the Cauchy–Schwarz relation is noted. Selected applications reveal the efficacy.Quanta 2019; 8: 36-43.