Cauchy-Schwarz inequality
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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
- <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 (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
- <math> |\langle x,y\rangle| \leq \|x\| \cdot \|y\|.\, </math>
The finite dimensional case of this inequality was proved by Cauchy in 1821, and the proof of general case was published by Bunyakovsky in 1859. The works of Schwarz appeared only 25 years later.
Proof
As the inequality is trivially true in the case y = 0, we may assume <y, y> is nonzero. Let <math> \lambda</math> be a complex number. Then,
- <math> 0 \leq \left\| x-\lambda y \right\|^2
= \langle x-\lambda y,x-\lambda y \rangle</math>
- <math> = \langle x,x \rangle - \overline\lambda \langle x,y \rangle - \lambda \langle y,x \rangle + |\lambda|^2 \langle y,y\rangle. </math>
Choosing
- <math> \lambda = \langle x,y \rangle \cdot \langle y,y \rangle^{-1}</math>
we obtain
- <math> 0 \leq \langle x,x \rangle - |\langle x,y \rangle|^2 \cdot \langle y,y \rangle^{-1}</math>
which is true if and only if
- <math> |\langle x,y \rangle|^2 \leq \langle x,x \rangle \cdot \langle y,y \rangle </math>
or equivalently:
- <math> \big| \langle x,y \rangle \big|
\leq \left\|x\right\| \left\|y\right\|. </math> Q.E.D.
Notable special cases
- In the case of the Euclidean space Rn, we get
- <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> 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: <math>|\mathbf{x} \cdot \mathbf{y}| = |\mathbf{x}| |\mathbf{y}| |\cos \theta| \le |\mathbf{x}| |\mathbf{y}|</math>. Also, the case can be deduced from Lagrange's identity by omitting a term.
- For the inner product space of square-integrable complex-valued functions, one has
- <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>
A generalization of these two inequalities is 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>
Usage
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\|^2</math> <math>= \langle x + y, x + y \rangle</math> x\|^2 + \langle x, y \rangle + \langle y, x \rangle + \|y\|^2</math> x\|^2 + 2|\langle x, y \rangle| + \|y\|^2</math> x\|^2 + 2\|x\|\|y\| + \|y\|^2</math> x\| + \|y\|\right)^2</math>
Taking the square roots gives the triangle inequality.
The Cauchy-Schwarz inequality is usually used to show Bessel's inequality.de:Cauchy-Schwarzsche Ungleichung es:Desigualdad de Cauchy-Schwarz fr:Inégalité de Cauchy-Schwarz it:Disuguaglianza di Cauchy-Schwarz he:אי שוויון קושי-שוורץ hu:Cauchy–Bunyakovszkij–Schwarz-egyenlőtlenség nl:Ongelijkheid van Cauchy-Schwarz ja:コーシー・シュワルツの不等式 no:Cauchy–Schwarz' ulikhet pl:Nierówność Schwarza ru:Неравенство Коши — Буняковского fi:Cauchyn epäyhtälö sv:Cauchy-Schwarz olikhet zh:柯西—施瓦茨不等式