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
An important consequence of the Cauchy-Schwarz inequality is that the inner product is a continuous function.
Formulated for Euclidean space Rn, we get
In the case of square-integrable complex-valued functions, we get
These latter two are generalized by the Hölder inequality.
The two sides are equal if and only if x and y are linearly dependent.