# Cauchy-Schwarz inequality

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*>

The two sides are equal if and only if

*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})

In the case of

square-integrable complex-valued

functionss, we get

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

These latter two are generalized by the Hölder inequality.

See also Triangle inequality.