Cauchy-Schwarz inequality

In mathematics, the Cauchy-Schwarz inequality, also known as the Schwarz inequality, or the Cauchy-Bunyakovski-Schwarz inequality, named after Augustin Louis Cauchy, Hermann Amandus Schwarz and Viktor Yakovlevich Bunyakovsky, is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in analysis applied to infinite series and integration of products, and in probability theory, applied to variances and covariances. The inequality states that if x and y are elements of real or complex inner product spaces then

Related Topics:
Mathematics - Augustin Louis Cauchy - Hermann Amandus Schwarz - Linear algebra - Vectors - Analysis - Infinite series - Integration - Probability theory - Variance - Covariance - Real - Complex - Inner product spaces

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:|langle x,y angle|^2 leq langle x,x angle cdot langle y,y angle.

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The two sides are equal if and only if x and y are linearly dependent.

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An important consequence of the Cauchy-Schwarz inequality is that the inner product is a continuous function.

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Another form of the Cauchy-Schwarz inequality is given using the notation of norm, as explained under norms on inner product spaces, as

Related Topics:
Norm - Norms on inner product spaces

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: |langle x,y angle| leq |x| cdot |y|.,

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