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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

( ∑ xi yi )2 ≤ ( ∑ xi2) · ( ∑ yi2)

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.



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