| # Cauchy-Schwartz Inequality For any two vectors , we have The above inequality is an equality if and only if are collinear. In other words: with optimal given by if is non-zero. Proof: The inequality is trivial if either one of the vectors is zero. Let us assume both are non-zero. Without loss of generality, we may re-scale and assume it has unit Euclidean norm (). Let us first prove that We consider the polynomial Since it is non-negative for every value of , its discriminant is non-positive. The Cauchy-Schwartz inequality follows. The second result is proven as follows. Let be the optimal value of the problem. The Cauchy-Schwartz inequality implies that . To prove that the value is attained (it is equal to its upper bound), we observe that if , then The vector is feasible for the optimization problem . This establishes a lower bound on the value of , :