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| {{more footnotes|date=July 2011}}
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| In [[mathematics]], the '''Cauchy–Schwarz inequality''' (the name of '''Bunyakovsky''' is sometimes added), is a useful [[Inequality (mathematics)|inequality]] encountered in many different settings, such as [[linear algebra]], [[mathematical analysis|analysis]], [[probability theory]], and other areas. It is considered to be one of the most important inequalities in all of mathematics.<ref name="Steele">[http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/CSMC_index.html The Cauchy–Schwarz Master Class: an Introduction to the Art of Mathematical Inequalities, Ch. 1] by [[J. Michael Steele]].</ref> It has a number of generalizations, among them [[Hölder's inequality]].
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| The inequality for sums was published by {{harvs|first=Augustin-Louis|last=Cauchy|authorlink=Augustin-Louis Cauchy|year=1821|txt=yes}}, while the corresponding inequality for integrals was first proved by
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| {{harvs|txt=yes|authorlink=Viktor Bunyakovsky|year=1859|first=Viktor|last=Bunyakovsky}}. The modern proof of the integral inequality was given by {{harvs|txt=yes|authorlink=Hermann Amandus Schwarz|first=Hermann Amandus|last=Schwarz|year=1888}}.<ref name="Steele" />
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| == Statement of the inequality ==
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| {{unreferenced section|date=July 2012}}
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| The Cauchy–Schwarz inequality states that for all vectors ''x'' and ''y'' of an [[inner product space]] it is true that
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| :<math> |\langle x,y\rangle| ^2 \leq \langle x,x\rangle \cdot \langle y,y\rangle,</math>
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| where <math>\langle\cdot,\cdot\rangle</math> is the [[inner product]] also known as dot product. Equivalently, by taking the square root of both sides, and referring to the [[inner product space#Norms on inner product spaces|norms]] of the vectors, the inequality is written as
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| : <math> |\langle x,y\rangle| \leq \|x\| \cdot \|y\|.\, </math>
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| Moreover, the two sides are equal if and only if ''x'' and ''y'' are [[linear independence|linearly dependent]] (or, in a geometrical sense, they are [[parallel (geometry)|parallel]] or one of the vectors' magnitude is zero).
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| If <math>x_1,\ldots, x_n\in\mathbb C</math> and <math>y_1,\ldots, y_n\in\mathbb C</math> have an imaginary component, the inner product is the standard inner product and the bar notation is used for complex conjugation then the inequality may be restated more explicitly as
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| :<math>|x_1 \bar{y}_1 + \cdots + x_n \bar{y}_n|^2 \leq (|x_1|^2 + \cdots + |x_n|^2) (|y_1|^2 + \cdots + |y_n|^2).</math>
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| When viewed in this way the numbers ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>, and ''y''<sub>1</sub>, ..., ''y''<sub>''n''</sub> are the components of ''x'' and ''y'' with respect to an [[orthonormal basis]] of ''V''.
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| Even more compactly written:
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| :<math>\left| \sum_{i=1}^n x_i \bar{y}_i \right|^2 \leq \sum_{j=1}^n |x_j|^2 \sum_{k=1}^n |y_k|^2 .</math>
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| Equality holds if and only if ''x'' and ''y'' are [[linearly dependent]], that is, one is a scalar multiple of the other (which includes the case when one or both are zero).
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| The finite-dimensional case of this inequality for real vectors was proven by Cauchy in 1821, and in 1859 Cauchy's student [[Viktor Bunyakovsky|Bunyakovsky]] noted that by taking limits one can obtain an integral form of Cauchy's inequality. The general result for an inner product space was obtained by [[Hermann Amandus Schwarz|Schwarz]] in the year 1888.
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| ==Proof==
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| Let ''u'', ''v'' be arbitrary vectors in a vector space ''V'' over ''F'' with an inner product, where ''F'' is the field of real or complex numbers. We prove the inequality
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| :<math> \big| \langle u,v \rangle \big|
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| \leq \left\|u\right\| \left\|v\right\|, \, </math>
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| and the fact that equality holds only when ''u'' and ''v'' are linearly dependent (the fact that conversely one has equality if ''u'' and ''v'' are linearly dependent is immediate from the properties of the inner product).
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| If {{nowrap|''v'' {{=}} 0}} it is clear that we have equality, and in this case ''u'' and ''v'' are also linearly dependent (regardless of ''u''). We henceforth assume that ''v'' is nonzero. Let
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| :<math>z= u-\frac {\langle u, v \rangle} {\langle v, v \rangle} v.</math>
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| Then, by linearity of the inner product in its first argument, one has
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| :<math>\langle z, v \rangle = \left\langle u -\frac {\langle u, v \rangle} {\langle v, v \rangle} v, v\right\rangle = \langle u, v \rangle - \frac {\langle u, v \rangle} {\langle v, v \rangle} \langle v, v \rangle = 0,</math>
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| i.e., ''z'' is a vector orthogonal to the vector ''v'' (Indeed, ''z'' is the [[vector projection|projection]] of ''u'' onto the plane orthogonal to ''v''.) We can thus apply the [[Pythagorean theorem#Inner product spaces|Pythagorean theorem]] to
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| :<math>u= \frac {\langle u, v \rangle} {\langle v, v \rangle} v+z,</math>
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| which gives
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| :<math>\left\|u\right\|^2 = \left|\frac{\langle u, v \rangle}{\langle v, v \rangle}\right|^2 \left\|v\right\|^2 + \left\|z\right\|^2 = \frac{|\langle u, v \rangle|^2}{\left\|v\right\|^2} + \left\|z\right\|^2 \geq \frac{|\langle u, v \rangle|^2}{\left\|v\right\|^2},</math>
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| and, after multiplication by ||''v''||<sup>2</sup>, the Cauchy–Schwarz inequality.
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| Moreover, if the relation '≥' in the above expression is actually an equality, then {{nowrap|{{!!}}''z''{{!!}}<sup>2</sup> {{=}} 0}} and hence {{nowrap|''z'' {{=}} 0}}; the definition of ''z'' then establishes a relation of linear dependence between ''u'' and ''v''. This establishes the theorem.
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| ==Special cases==
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| === R<sup>''n''</sup> ===
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| In [[Euclidean space]] <math> \mathbb R ^n </math> with the standard inner product, the Cauchy–Schwarz inequality is
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| :<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>
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| To prove this form of the inequality, consider the following quadratic polynomial in ''z''.
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| :<math>(x_1 z + y_1)^2 + \cdots + (x_n z + y_n)^2 = \left( \sum ( x_i^2) \right) \cdot z^2 + 2 \cdot \left( \sum ( x_i \cdot y_i) \right) \cdot z + \sum ( y_i^2) </math>
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| Since it is nonnegative it has at most one real root in ''z'', whence its [[discriminant]] is less than or equal to zero, that is,
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| :<math>\left(\sum ( x_i \cdot y_i ) \right)^2 - \sum {x_i^2} \cdot \sum {y_i^2} \le 0,</math>
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| which yields the Cauchy–Schwarz inequality.
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| An equivalent proof for <math> \mathbb R ^n </math> starts with the summation below.
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| Expanding the brackets we have:
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| :<math> \sum_{i=1}^n \sum_{j=1}^n \left( x_i y_j - x_j y_i \right)^2
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| = \sum_{i=1}^n x_i^2 \sum_{j=1}^n y_j^2 + \sum_{j=1}^n x_j^2 \sum_{i=1}^n y_i^2
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| - 2 \sum_{i=1}^n x_i y_i \sum_{j=1}^n x_j y_j </math>,
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| collecting together identical terms (albeit with different summation indices) we find:
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| :<math> \frac{1}{2} \sum_{i=1}^n \sum_{j=1}^n \left( x_i y_j - x_j y_i \right)^2
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| = \sum_{i=1}^n x_i^2 \sum_{i=1}^n y_i^2 - \left( \sum_{i=1}^n x_i y_i \right)^2 . </math>
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| Because the left-hand side of the equation is a sum of the squares of real numbers it is greater than or equal to zero, thus:
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| :<math>
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| \sum_{i=1}^n x_i^2 \sum_{i=1}^n y_i^2 - \left( \sum_{i=1}^n x_i y_i \right)^2 \geq 0.
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| </math>
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| Yet another approach when ''n'' ≥ 2 (''n'' = 1 is trivial) is to consider the plane containing ''x'' and ''y''. More precisely, recoordinatize R<sup>''n''</sup> with any [[orthonormal]] basis whose first two vectors span a subspace containing ''x'' and ''y''. In this basis only <math>x_1,~x_2,~y_1</math> and <math>y_2~</math> are nonzero, and the inequality reduces to the algebra of dot product in the plane, which is related to the angle between two vectors, from which we obtain the inequality:
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| :<math>|x \cdot y| = \|x\| \|y\| | \cos \theta | \le \|x\| \|y\|.</math>
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| When ''n'' = 3 the Cauchy–Schwarz inequality can also be deduced from [[Lagrange's identity]], which takes the form
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| :<math>\langle x,x\rangle \cdot \langle y,y\rangle = |\langle x,y\rangle|^2 + |x \times y|^2</math>
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| from which readily follows the Cauchy–Schwarz inequality.
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| Another proof of the general case for n can be done by using the technique used to prove [[Inequality of arithmetic and geometric means#Proof by induction using basic calculus|Inequality of arithmetic and geometric means]].
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| ===L<sup>2</sup>===
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| For the inner product space of [[square-integrable]] complex-valued [[function (mathematics)|functions]], one has
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| :<math>\left|\int_{\mathbb{R}^n} f(x) \overline{g(x)}\,dx\right|^2\leq\int_{\mathbb{R}^n} \left|f(x)\right|^2\,dx \cdot \int_{\mathbb{R}^n}\left|g(x)\right|^2\,dx.</math>
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| A generalization of this is the [[Hölder inequality]].
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| == Applications ==
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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'':
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| : <math>
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| \begin{align}
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| \|x + y\|^2 & = \langle x + y, x + y \rangle \\
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| & = \|x\|^2 + \langle x, y \rangle + \langle y, x \rangle + \|y\|^2 \\
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| & = \|x\|^2 + 2 \text{ Re} \langle x, y \rangle + \|y\|^2\\
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| & \le \|x\|^2 + 2|\langle x, y \rangle| + \|y\|^2 \\
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| & \le \|x\|^2 + 2\|x\|\|y\| + \|y\|^2 \\
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| & = \left (\|x\| + \|y\|\right)^2.
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| \end{align}
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| </math>
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| Taking square roots gives the triangle inequality.
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| The Cauchy–Schwarz inequality allows one to extend the notion of "angle between two vectors" to any [[real numbers|real]] inner product space, by defining:
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| :<math>
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| \cos\theta_{xy}=\frac{\langle x,y\rangle}{\|x\| \|y\|}.
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| </math>
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| The Cauchy–Schwarz inequality proves that this definition is sensible, by showing that the right-hand side lies in the interval [−1, 1], and justifies the notion that (real) Hilbert spaces are simply generalizations of the Euclidean space.
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| It can also be used to define an angle in [[complex numbers|complex]] [[inner product space]]s, by taking the absolute value of the right-hand side, as is done when extracting a metric from [[Fidelity of quantum states|quantum fidelity]].
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| The Cauchy–Schwarz is used to prove that the inner product is a [[continuous function]] with respect to the [[topology]] induced by the inner product itself.
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| The Cauchy–Schwarz inequality is usually used to show [[Bessel's inequality]].
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| ===Probability theory===
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| For the multivariate case,{{clarify|reason=define GE operator here|date=July 2011}} <math>\text{Var}\left(Y\right)\ge\text{Cov}\left(Y,X\right)\text{Var}^{-1}\left(X\right)\text{Cov}\left(X,Y\right) .</math>.<ref>{{cite journal|last=Gautam|first=Tripathi|title=A matrix extension of the Cauchy-Schwarz inequality|journal=Economics Letters|date=4 December 1998|url=http://web2.uconn.edu/tripathi/published-papers/cs.pdf}}</ref> This inequality means that the diference is semidefinite positive.
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| For the univariate case, <math>\text{Var}\left(Y\right)\ge\frac{\text{Cov}\left(Y,X\right)\text{Cov}\left(Y,X\right)}{\text{Var}\left(X\right)}.</math> Indeed, for [[random variable]]s ''X'' and ''Y'', the expectation of their product is an inner product. That is,
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| :<math>\langle X, Y \rangle \triangleq \operatorname{E}(X Y),</math>
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| and so, by the Cauchy–Schwarz inequality,
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| :<math>|\operatorname{E}(XY)|^2 \leq \operatorname{E}(X^2) \operatorname{E}(Y^2).</math>
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| Moreover, if ''μ'' = E(''X'') and ''ν'' = E(''Y''), then
| |
| | |
| :<math> \begin{align}
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| |\operatorname{Cov}(X,Y)|^2
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| &= |\operatorname{E}( (X - \mu)(Y - \nu) )|^2 = | \langle X - \mu, Y - \nu \rangle |^2\\
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| &\leq \langle X - \mu, X - \mu \rangle \langle Y - \nu, Y - \nu \rangle \\
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| & = \operatorname{E}( (X-\mu)^2 ) \operatorname{E}( (Y-\nu)^2 ) \\
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| & = \operatorname{Var}(X) \operatorname{Var}(Y),
| |
| \end{align}</math>
| |
| | |
| where Var denotes [[variance]] and Cov denotes [[covariance]].
| |
| | |
| == Generalizations ==
| |
| Various generalizations of the Cauchy–Schwarz inequality exist in the context of [[operator theory]], e.g. for operator-convex functions, and [[operator algebra]]s, where the domain and/or range of ''φ'' are replaced by a [[C*-algebra]] or [[W*-algebra]].
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| This section lists a few of such inequalities from the operator algebra setting, to give a flavor of results of this type.
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| === Positive functionals on C*- and W*-algebras ===
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| One can discuss inner products as positive functionals. Given a Hilbert space ''L''<sup>2</sup>(''m''), ''m'' being a finite measure, the inner product < · , · > gives rise to a positive functional ''φ'' by
| |
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| :<math>\phi (g) = \langle g, 1 \rangle.</math>
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| Since < ''ƒ'', ''ƒ'' > ≥ 0, ''φ''(''f*f'') ≥ 0 for all ''ƒ'' in ''L''<sup>2</sup>(''m''), where ''ƒ*'' is pointwise conjugate of ''ƒ''. So ''φ'' is positive. Conversely every positive functional ''φ'' gives a corresponding inner product < ''ƒ'', ''g'' ><sub>''φ''</sub> = ''φ''(''g*ƒ''). In this language, the Cauchy–Schwarz inequality becomes
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| :<math>| \phi(g^*f) |^2 \leq \phi(f^*f) \phi(g^*g), \, </math>
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| | |
| which extends verbatim to positive functionals on C*-algebras.
| |
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| We now give an operator theoretic proof for the Cauchy–Schwarz inequality which passes to the C*-algebra setting. One can see from the proof that the Cauchy–Schwarz inequality is a consequence of the ''positivity'' and ''anti-symmetry'' inner-product axioms.
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| Consider the positive matrix
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| | |
| :<math>
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| M =
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| \begin{bmatrix}
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| f^*\\
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| g^*
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| \end{bmatrix}
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| \begin{bmatrix}
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| f & g
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| f^*f & f^* g \\
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| g^*f & g^*g
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| \end{bmatrix}.
| |
| </math>
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| | |
| Since ''φ'' is a positive linear map whose range, the complex numbers '''C''', is a commutative C*-algebra, ''φ'' is [[completely positive map|completely positive]]. Therefore
| |
| | |
| :<math>
| |
| M' = (I_2 \otimes \phi)(M) =
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| \begin{bmatrix}
| |
| \phi(f^*f) & \phi(f^* g) \\
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| \phi(g^*f) & \phi(g^*g)
| |
| \end{bmatrix}
| |
| </math>
| |
| | |
| is a positive 2 × 2 scalar matrix, which implies it has positive determinant:
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| | |
| :<math>
| |
| \phi(f^*f) \phi(g^*g) - | \phi(g^*f) |^2 \geq 0 \quad \text{i.e.} \quad \phi(f^*f) \phi(g^*g) \geq | \phi(g^*f) |^2. \,
| |
| </math> | |
| | |
| This is precisely the Cauchy–Schwarz inequality. If ''ƒ'' and ''g'' are elements of a C*-algebra, ''f*'' and ''g*'' denote their respective adjoints.
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| We can also deduce from above that every positive linear functional is bounded, corresponding to the fact that the inner product is jointly continuous.
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| === Positive maps ===
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| Positive functionals are special cases of [[Choi's theorem on completely positive maps|positive map]]s. A linear map Φ between C*-algebras is said to be a '''positive map''' if ''a'' ≥ 0 implies Φ(''a'') ≥ 0. It is natural to ask whether inequalities of Schwarz-type exist for positive maps. In this more general setting, usually additional assumptions are needed to obtain such results.
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| | |
| ==== Kadison–Schwarz inequality ====
| |
| The following theorem is named after [[Richard Kadison]].
| |
| | |
| '''Theorem.''' If Φ is a unital positive map, then for every [[normal operator|normal element]] ''a'' in its domain, we have Φ(''a*a'') ≥ Φ(''a*'')Φ(''a'') and Φ(''a*a'') ≥ Φ(''a'')Φ(''a*'').
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| | |
| This extends the fact ''φ''(''a*a'') · 1 ≥ ''φ''(''a'')*''φ''(''a'') = |''φ''(''a'')|<sup>2</sup>, when ''φ'' is a linear functional.
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| The case when ''a'' is self-adjoint, i.e. ''a = a*'', is sometimes known as '''Kadison's inequality'''.
| |
| | |
| ==== 2-positive maps ====
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| When Φ is 2-positive, a stronger assumption than merely positive, one has something that looks very similar to the original Cauchy–Schwarz inequality:
| |
| | |
| '''Theorem''' (''Modified Schwarz inequality for 2-positive maps'')<ref>{{Citation|last1=Paulsen|url=http://books.google.com/books?id=VtSFHDABxMIC&pg=PA40|title=Completely Bounded Maps and Operator Algebras|isbn=9780521816694|year=2002}} page 40.</ref> For a 2-positive map Φ between C*-algebras, for all ''a'', ''b'' in its domain,
| |
| | |
| # <math>\Phi(a)^*\Phi(a) \leq \Vert\Phi(1)\Vert\Phi(a^*a)</math>
| |
| # <math>\Vert\Phi(a^*b)\Vert^2 \leq \Vert\Phi(a^*a)\Vert \cdot \Vert\Phi(b^*b)\Vert.</math>
| |
| | |
| A simple argument for (2) is as follows. Consider the positive matrix
| |
| | |
| :<math>
| |
| M=
| |
| \begin{bmatrix}
| |
| a^* & 0 \\
| |
| b^* & 0
| |
| \end{bmatrix}
| |
| \begin{bmatrix}
| |
| a & b \\
| |
| 0 & 0
| |
| \end{bmatrix}
| |
| =
| |
| \begin{bmatrix}
| |
| a^*a & a^* b \\
| |
| b^*a & b^*b
| |
| \end{bmatrix}.
| |
| </math>
| |
| | |
| By 2-positivity of Φ,
| |
| | |
| :<math>
| |
| (I_2 \otimes \Phi) M =
| |
| \begin{bmatrix}
| |
| \Phi(a^*a) & \Phi(a^* b) \\
| |
| \Phi(b^*a) & \Phi(b^*b)
| |
| \end{bmatrix}
| |
| </math>
| |
| | |
| is positive. The desired inequality then follows from the properties of positive 2 × 2 (operator) matrices.
| |
| | |
| Part (1) is analogous. One can replace the matrix <math>\begin{bmatrix} a & b \\ 0 & 0 \end{bmatrix}</math> by <math>\begin{bmatrix} 1 & a \\ 0 & 0 \end{bmatrix}.</math>
| |
| | |
| ==Reforming Cauchy-Schwarz Inequality for cross product==
| |
| | |
| The same applies to the cross product space (where u and v are not the zero vector):
| |
| | |
| : <math> \| \mathbf{v} \times \mathbf{u} \| \leq \|v\| \cdot \|u\|.\, </math>
| |
| | |
| | |
| ===Proof===
| |
| | |
| : <math> \| \mathbf{v} \times \mathbf{u} \| \leq \|v\| \cdot \|u\|.\, </math>
| |
| | |
| write the outer product magnitude as <math> \| \mathbf{v} \times \mathbf{u} \| = \|v\| \cdot \| u \| \cdot \sin{(\mathbf{v} , \mathbf{u})} </math>
| |
| | |
| The magnitudes are always positive so we can multiply the inequality with <math> \frac{1}{\|v\| \cdot \| u \|} </math>
| |
| | |
| The magnitudes cancel and we finally get
| |
| | |
| : <math> | \sin{(\mathbf{v} , \mathbf{u})} | \leq 1 </math>
| |
| | |
| which is always true since the sine function <math> f((\mathbf{v} , \mathbf{u})) = \sin{(\mathbf{v} , \mathbf{u})} </math> is a function <math> f: \mathbb{R} \to [-1,1] </math>
| |
| | |
| == Physics ==
| |
| The general formulation of the [[uncertainty principle|Heisenberg uncertainty principle]] is derived using the Cauchy–Schwarz inequality in the [[Hilbert space]] of [[observable|quantum observables]].
| |
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| == See also ==
| |
| * [[Hölder's inequality]]
| |
| * [[Minkowski inequality]]
| |
| * [[Jensen's inequality]]
| |
| | |
| ==Notes==
| |
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| {{Reflist}}
| |
| | |
| ==References==
| |
| {{refbegin}}
| |
| *{{springer|id=b/b017770|title=Bunyakovskii inequality|first=V.I. |last=Bityutskov}}
| |
| *{{citation|first=V. |last= Bouniakowsky |authorlink=Viktor Yakovlevich Bunyakovsky |title= Sur quelques inegalités concernant les intégrales aux différences finies|journal= Mem. Acad. Sci. St. Petersbourg |volume=7|issue= 1 |year=1859|pages= 9|url=http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/bunyakovsky.pdf|format=PDF}}
| |
| *{{citation|first=A. |last=Cauchy|title= Oeuvres 2, III|page=373|year=1821}}
| |
| *{{citation|first=S. S. |last=Dragomir|title=A survey on Cauchy–Bunyakovsky–Schwarz type discrete inequalities|journal=JIPAM. J. Inequal. Pure Appl. Math.|volume=4|issue=3|year=2003|pages=142 pp|url=http://jipam.vu.edu.au/article.php?sid=301}}
| |
| *{{citation|first=R.V.|last= Kadison|authorlink=Richard V. Kadison|title= A generalized Schwarz inequality and algebraic invariants for operator algebras|journal=Annals of Mathematics|volume=56|year= 1952|doi=10.2307/1969657|pages=494–503|jstor=1969657|issue=3}}.
| |
| *{{Citation|last=Lohwater|first=Arthur|title=Introduction to Inequalities|publisher=Online e-book in PDF fomat|url=http://www.mediafire.com/?1mw1tkgozzu|year=1982|isbn=}}
| |
| *{{citation|first=V. |last=Paulsen|title=Completely Bounded Maps and Operator Algebras|publisher= Cambridge University Press|year= 2003}}.
| |
| *{{citation|first=H. A. |last=Schwarz|year=1888 |pages=318|journal=Acta Societatis scientiarum Fennicae|volume=XV |title=Über ein Flächen kleinsten Flächeninhalts betreffendes Problem der Variationsrechnung|url=http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/Schwarz.pdf|format=PDF}}
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| *{{springer|title=Cauchy inequality|id=C/c020880|first=E.D. |last=Solomentsev}}
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| *{{citation|url=http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/CSMC_index.html |first=J.M. |last=Steele|title=The Cauchy–Schwarz Master Class|publisher= Cambridge University Press|year=2004|isbn=0-521-54677-X}}
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| {{refend}}
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| ==External links==
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| * [http://jeff560.tripod.com/c.html Earliest Uses: The entry on the Cauchy–Schwarz inequality has some historical information.]
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| * [http://people.revoledu.com/kardi/tutorial/LinearAlgebra/LinearlyIndependent.html#LinearlyIndependentVectors Example of application of Cauchy–Schwarz inequality to determine Linearly Independent Vectors] Tutorial and Interactive program.
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| {{Functional Analysis}}
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| {{DEFAULTSORT:Cauchy-Schwarz inequality}}
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| [[Category:Inequalities]]
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| [[Category:Linear algebra]]
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| [[Category:Operator theory]]
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| [[Category:Articles containing proofs]]
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| [[Category:Probability theory]]
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| [[Category:Mathematical analysis]]
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