In mathematics, the Cauchy-Schwarz inequality, also known as the Schwarz inequality, the Cauchy inequality, or the Cauchy-Bunyakovski-Schwarz inequality, named after Augustin Louis Cauchy, Viktor Yakovlevich Bunyakovsky and Hermann Amandus Schwarz, 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

$|\backslash langle\; x,y\backslash rangle|^2\; \backslash leq\; \backslash langle\; x,x\backslash rangle\; \backslash cdot\; \backslash langle\; y,y\backslash rangle.$

The two sides are equal if and only if x and y are linearly dependent (or in geometrical sense they are parallel). This contrasts with a property that the inner product of two vectors is zero if they are orthogonal (or perpendicular) to each other.

The inequality hence confers the notion of "the angle between the two vectors" to an inner product, where concepts of Euclidean geometry may not have meaningful sense, and justifies that the notion that inner product spaces are generalizations of Euclidean space.

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

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

$|\backslash langle\; x,y\backslash rangle|\; \backslash leq\; \backslash |x\backslash |\; \backslash cdot\; \backslash |y\backslash |.\backslash ,$

The finite dimensional case of this inequality was proved by Cauchy in 1821, and the proof of the general case was published by Bunyakovsky in 1859. The works of Schwarz appeared only 25 years later.

Proof

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As the inequality is trivially true in the case y = 0, we may assume <y, y> is nonzero. Let $\backslash lambda$ be a complex number. Then,

$0\; \backslash leq\; \backslash left\backslash |\; x-\backslash lambda\; y\; \backslash right\backslash |^2$

= \langle x-\lambda y,x-\lambda y \rangle

$=\; \backslash langle\; x,x\; \backslash rangle\; -\; \backslash lambda\; \backslash langle\; x,y\; \backslash rangle\; -\; \backslash lambda\; \backslash langle\; y,x\; \backslash rangle\; +\; |\backslash lambda|^2\; \backslash langle\; y,y\backslash rangle.$

Choosing

$\backslash lambda\; =\; \backslash langle\; x,y\; \backslash rangle\; \backslash cdot\; \backslash langle\; y,y\; \backslash rangle^\{-1\}$

we obtain

$0\; \backslash leq\; \backslash langle\; x,x\; \backslash rangle\; -\; |\backslash langle\; x,y\; \backslash rangle|^2\; \backslash cdot\; \backslash langle\; y,y\; \backslash rangle^\{-1\}$

which is true if and only if

$|\backslash langle\; x,y\; \backslash rangle|^2\; \backslash leq\; \backslash langle\; x,x\; \backslash rangle\; \backslash cdot\; \backslash langle\; y,y\; \backslash rangle$

or equivalently:

$\backslash big|\; \backslash langle\; x,y\; \backslash rangle\; \backslash big|$

\leq \left\|x\right\| \left\|y\right\|. Q.E.D.

Notable special cases

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• In the case of the Euclidean space Rⁿ, we get

$\backslash left(\backslash sum\_\{i=1\}^n\; x\_i\; y\_i\backslash right)^2\backslash leq\; \backslash left(\backslash sum\_\{i=1\}^n\; x\_i^2\backslash right)\; \backslash left(\backslash sum\_\{i=1\}^n\; y\_i^2\backslash right).$ In particular, in the Euclidean space of 2 or 3, if the dot product is defined in terms of the angle between two vectors, then one can immediately see the inequality: $|\backslash mathbf\{x\}\; \backslash cdot\; \backslash mathbf\{y\}|\; =\; |\backslash mathbf\{x\}|\; |\backslash mathbf\{y\}|\; |\backslash cos\; \backslash theta|\; \backslash le\; |\backslash mathbf\{x\}|\; |\backslash mathbf\{y\}|$. Also, in this case the Cauchy-Schwarz inequality can be deduced from Lagrange's identity by omitting a term. In three dimensions n=3, Lagrange's identity takes the form

$\backslash langle\; x,x\backslash rangle\; \backslash cdot\; \backslash langle\; y,y\backslash rangle\; =\; |\backslash langle\; x,y\backslash rangle|^2\; +\; |x\; \backslash times\; y|^2.$

• For the inner product space of square-integrable complex-valued functions, one has

$\backslash left|\backslash int\; f(x)g(x)\backslash ,dx\backslash right|^2\backslash leq\backslash int\; \backslash left|f(x)\backslash right|^2\backslash ,dx\; \backslash cdot\; \backslash int\backslash left|g(x)\backslash right|^2\backslash ,dx.$

A generalization of these two inequalities is the Hölder inequality.

Usage

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The triangle inequality for the inner product is often shown as a consequence of the Cauchy-Schwarz inequality, as follows: given vectors x and y,

{| \ >x + y\ $=\; \backslash langle\; x\; +\; y,\; x\; +\; y\; \backslash rangle$ = \ >x\ \le \ >x\ \le \ >x\ = \left(\ >x\

Taking the square roots gives the triangle inequality.

The Cauchy-Schwarz inequality is usually used to show Bessel's inequality.

Inequalities

Cauchy-Schwarz' ulighed | Cauchy-Schwarzsche Ungleichung | Desigualdad de Cauchy-Schwarz | Inégalité de Cauchy-Schwarz | Disuguaglianza di Cauchy-Schwarz | אי שוויון קושי-שוורץ | Cauchy–Bunyakovszkij–Schwarz-egyenlőtlenség | Ongelijkheid van Cauchy-Schwarz | コーシー・シュワルツの不等式 | Cauchy–Schwarz' ulikhet | Nierówność Schwarza | Desigualdade de Cauchy-Schwarz | Неравенство Коши — Буняковского | Cauchyn epäyhtälö | Cauchy-Schwarz olikhet | 柯西—施瓦茨不等式