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 talking about vectors, and in analysis talking about infinite series and integration of products. The inequality states that if

*x*and

*y*are elements of a real or complex inner product spaces then

- |<
*x*,*y*>|^{2}≤ <*x*,*x*> · <*y*,*y*>

*x*and

*y*are linearly dependent.

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

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

- ( ∑
*x*_{i}*y*_{i})^{2}≤ ( ∑*x*_{i}^{2}) · ( ∑*y*_{i}^{2})

- | ∫
*f*^{ *}*g*d*x*|^{2}≤ ( ∫ |*f*|^{2}d*x*) · ( ∫ |*g*|^{2}d*x*)

See also Triangle inequality.