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>
An important consequence of the Cauchy-Schwarz inequality is that the inner product is a continuous function.
Formulated for Euclidean space Rn, we get
- ( ∑ xi yi )2 ≤ ( ∑ xi2) · ( ∑ yi2)
- | ∫ f * g dx|2 ≤ ( ∫ |f|2 dx) · ( ∫ |g|2 dx)
See also Triangle inequality.