Cauchy-Schwarz inequality

From Academic Kids

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

<math>|\langle x,y\rangle|^2 \leq \langle x,x\rangle \cdot \langle y,y\rangle.<math>

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.

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

<math> |\langle x,y\rangle| \leq \|x\| \cdot \|y\|.\, <math>

Proof

  • Real inner product spaces

To prove this inequality note it is trivial in the case y = 0. Thus we may assume <y, y> is nonzero. Further, let <math> \lambda \in \mathbb{R} <math>. Thus we may let

<math> 0 \leq \langle x-\lambda y,x-\lambda y \rangle<math>
which equals
<math> = \langle x-\lambda y,x \rangle - \lambda \langle x-\lambda y,y \rangle. <math>
We now choose <math> \lambda = \langle y,x \rangle \cdot \|y\|^{-2}. <math>
Plugging in for <math> \lambda <math> we wind up getting
<math> 0 \leq \|x\| ^2 - \langle x,y \rangle^2 \cdot \|y\|^{-2}<math>
which is true <math> \iff <math>
<math> \langle x,y \rangle^2 \leq \|x\|^2 \|y\|^2. <math>
Taking the square root gives us
<math> \big| \langle x,y \rangle \big| \leq \|x\| \|y\| <math> Q.E.D.

Notable special cases

<math>\left(\sum_{i=1}^n x_i y_i\right)^2\leq \left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n y_i^2\right).<math>
<math>\left|\int f^*(x)g(x)\,dx\right|^2\leq\int \left|f(x)\right|^2\,dx \cdot \int\left|g(x)\right|^2\,dx.<math>

These latter two are generalized by the Hölder inequality.

  • A notable strengthening of the basic inequality occurs in dimension n = 3, where a stronger equality holds:
<math>\langle x,x\rangle \cdot \langle y,y\rangle = |\langle x,y\rangle|^2 + |x \times y|^2.<math>

See also

es:Desigualdad de Cauchy-Schwarz fr:Ingalit de Cauchy-Schwarz ja:コーシー・シュワルツの不等式 pl:Nierwność Schwarza ru:Неравенство Коши—Буняковского zh:柯西—施瓦茨不等式

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