# Cauchy-Schwarz inequality

In mathematics, the **Cauchy-Schwarz inequality**, also known as the **Schwarz inequality**, or the **Cauchy-Bunyakovski-Schwarz inequality**, is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in analysis applied to infinite series and integration of products, and in probability theory, applied to variances and covariances. The inequality states that if *x* and *y* are elements of real or complex inner product spaces then

*x*and

*y*are linearly dependent.

An important consequence of the Cauchy-Schwarz inequality is that the inner product is a continuous function.

## Notable special cases

Formulated for Euclidean space **R**^{n}, we get

See also triangle inequality.