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
of products. The inequality states that if x
are elements of a real
or complex inner product spaces
- |<x, y>|2 ≤ <x, x> · <y, y>
The two sides are equal if and only if x
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
- ( ∑ xi yi )2 ≤ ( ∑ xi2) · ( ∑ yi2)
In the case of square-integrable
, we get
- | ∫ f * g dx|2 ≤ ( ∫ |f|2 dx) · ( ∫ |g|2 dx)
These latter two are generalized by the Hölder inequality.
See also Triangle inequality.