The Cauchy–Schwarz inequality states that for all vectors x and y of an inner product space it is true that

 |\langle x,y\rangle| ^2 \leq \langle x,x\rangle \cdot \langle y,y\rangle,

where \langle\cdot,\cdot\rangle is the inner product. Equivalently, by taking the square root of both sides, and referring to the norms of the vectors, the inequality is written as

 |\langle x,y\rangle| \leq \|x\| \cdot \|y\|.\,

Notable special cases


In Euclidean space Rn with the standard inner product, the Cauchy–Schwarz inequality is

\left(\sum_{i=1}^n x_i y_i\right)^2\leq \left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n y_i^2\right).