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 Rⁿ, we get

( ∑ x_i y_i )² ≤ ( ∑ x_i²) · ( ∑ y_i²)

In the case of square-integrable complex-valued functions, we get

| ∫ f ^* g dx|² ≤ ( ∫ |f|² dx) · ( ∫ |g|² dx)

These latter two are generalized by the Hölder inequality.