Main Page | See live article | Alphabetical index 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 ( ∑ xi yi )2 ≤ ( ∑ xi2) · ( ∑ yi2) In the case of square-integrable complex-valued functionss, 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. This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.