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In [[mathematics]], the '''Cauchy–Schwarz inequality''', also known as the '''Cauchy–Bunyakovsky–Schwarz inequality''', is a useful [[Inequality (mathematics)|inequality]] encountered in many different settings, such as [[linear algebra]], [[mathematical analysis|analysis]], [[probability theory]], [[vector algebra]] and other areas. It is considered to be one of the most important inequalities in all of mathematics.<ref name="Steele">{{cite book |last=Steele |first=J. Michael |year=2004 |title=The Cauchy–Schwarz Master Class: an Introduction to the Art of Mathematical Inequalities |publisher=The Mathematical Association of America |isbn=978-0521546775 |page=1 |url=http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/CSMC_index.html |quote=...there is no doubt that this is one of the most widely used and most important inequalities in all of mathematics.}}</ref>

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
{{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" />

== Statement of the inequality ==
The Cauchy–Schwarz inequality states that for all vectors <math>u</math> and <math>v</math> of an [[inner product space]] it is true that
: <math>|\langle \mathbf{u},\mathbf{v}\rangle| ^2 \leq \langle \mathbf{u},\mathbf{u}\rangle \cdot \langle \mathbf{v},\mathbf{v}\rangle,</math>
where <math>\langle\cdot,\cdot\rangle</math> is the [[inner product]]. Examples of inner products include the real and complex [[dot product]]; see the [[Inner product space#Examples|examples in inner 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<ref name=Strang5>{{cite book |last=Strang |first=Gilbert |date=19 July 2005 |title=Linear Algebra and its Applications |edition=4th |chapter=3.2 |publisher=Cengage Learning |location=Stamford, CT |isbn=978-0030105678 |pages=154–155}}</ref><ref name=":0">{{cite book |last1=Hunter |first1=John K. |last2=Nachtergaele |first2=Bruno |year=2001 |title=Applied Analysis |publisher=World Scientific |isbn=981-02-4191-7 |url=https://books.google.com/books?id=oOYQVeHmNk4C}}</ref>
: <math>|\langle \mathbf{u},\mathbf{v}\rangle| \leq \|\mathbf{u}\|  \|\mathbf{v}\|.</math>
Moreover, the two sides are equal if and only if <math>\mathbf{u}</math> and <math>\mathbf{v}</math> are [[linear independence|linearly dependent]] (meaning they are [[parallel (geometry)|parallel]]: one of the vector's magnitudes is zero, or one is a scalar multiple of the other).<ref>{{cite book |last1=Bachmann |first1=George |last2=Narici |first2=Lawrence |last3=Beckenstein |first3=Edward |date=2012-12-06 |title=Fourier and Wavelet Analysis |publisher=Springer Science & Business Media |isbn=9781461205050 |page=14 |url=https://books.google.com/books?id=PkHhBwAAQBAJ}}</ref><ref>{{cite book |last=Hassani |first=Sadri |year=1999 |title=Mathematical Physics: A Modern Introduction to Its Foundations |publisher=Springer |isbn=0-387-98579-4 |page=29 |quote=Equality holds iff <c&#124;c>=0 or &#124;c>=0. From the definition of &#124;c>, we conclude that &#124;a> and &#124;b> must be proportional.}}</ref>

If <math>u_1,\ldots, u_n\in\mathbb C</math> and <math>v_1,\ldots, v_n\in\mathbb C</math>, and the inner product is the standard complex inner product, then the inequality may be restated more explicitly as follows (where the bar notation is used for [[Complex conjugate|complex conjugation]]): 
: <math>|u_1\bar{v}_1 + \cdots + u_n \bar{v}_n|^2 \leq (|u_1|^2 + \cdots + |u_n|^2) (|v_1|^2 + \cdots + |v_n|^2)</math>
or 
: <math>\left| \sum_{i=1}^n u_i \bar{v}_i \right|^2 \leq \sum_{j=1}^n |u_j|^2 \sum_{k=1}^n |v_k|^2.</math>

==Proofs==

=== First proof ===
Let <math>u</math> and <math>v</math> be arbitrary vectors in a vector space over <math>\mathbb F</math> with an inner product, where <math>\mathbb F</math> is the field of real or complex numbers. We prove the inequality
: <math>\big| \langle u,v \rangle \big| \leq \|u\| \|v\|</math>
and that equality holds if and only if either <math>u</math> or <math>v</math> is a multiple of the other (which includes the special case that either is the zero vector).

If <math>v = 0</math>, it is clear that there is equality, and in this case <math>u</math> and <math>v</math> are also linearly dependent, regardless of <math>u</math>, so the theorem is true. Similarly if <math>u = 0</math>. One henceforth assumes that <math>v</math> is nonzero.

Let
: <math>z= u-u_v= u-\frac {\langle u, v \rangle} {\langle v, v \rangle} v.</math>
Then, by linearity of the inner product in its first argument, one has
: <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>
Therefore,  <math>z</math> is a vector orthogonal to the vector <math>v</math> (Indeed, <math>z</math>  is the [[vector projection|projection]] of <math>u</math>  onto the plane orthogonal to <math>v</math> .) We can thus apply the [[Pythagorean theorem#Inner product spaces|Pythagorean theorem]] to
: <math>u= \frac {\langle u, v \rangle} {\langle v, v \rangle} v+z</math>
which gives
: <math>\|u\|^2 = \left|\frac{\langle u, v \rangle}{\langle v, v \rangle}\right|^2 \|v\|^2 + \|z\|^2 = \frac{|\langle u, v \rangle|^2}{( \|v\|^2 )^2}\,\|v\|^2 +\|z\|^2= \frac{|\langle u, v \rangle|^2}{\|v\|^2} + \|z\|^2 \geq \frac{|\langle u, v \rangle|^2}{\|v\|^2}</math>
and, after multiplication by <math>\| v \|^2</math> and taking square root, we get the Cauchy–Schwarz inequality.
Moreover, if the relation <math>\geq</math> in the above expression is actually an equality, then <math>\| z \|^2 = 0</math> and hence <math>z = 0</math>; the definition of <math>z</math> then establishes a relation of linear dependence between <math>u</math> and <math>v</math>. On the other hand, if <math>u</math> and <math>v</math> are linearly dependent, then there exists <math>c \in \mathbb{F}</math> such that <math>u=c \cdot v</math> (since <math>v \ne 0</math>). Then 
: <math>|\langle u, v \rangle| = |\langle c \cdot v, v \rangle| = \left| c\|v\|^2 \right| = |c|\|v\|^2=\|c \cdot v\|\|v\|=\|u\|\|v\|.</math>
This establishes the theorem.

=== Second proof ===
Let <math>u</math> and <math>v</math> be arbitrary vectors in an inner product space over <math>\mathbb C</math>.

In the special case <math>v = 0</math> the theorem is trivially true. Now assume that <math>v \ne 0</math>. Let <math> \lambda \in \mathbb{C}</math> be given by <math>\lambda = \langle u, v \rangle / \|v\|^2</math>, then

: <math>
\begin{align}
0 & \leq \| u - \lambda \cdot v \|^2 \\
 & = \langle u, u \rangle - \langle \lambda \cdot v, u \rangle - \langle u,\lambda \cdot v \rangle + \langle \lambda \cdot v, \lambda \cdot v \rangle   \\
 & =  \langle u, u \rangle - \lambda \langle v, u \rangle - \overline{\lambda} \langle u, v \rangle + \lambda \overline{\lambda} \langle v, v \rangle \\
 & =  \|u\|^2 - \lambda \overline{\langle u, v \rangle} - \overline{\lambda} \langle u, v \rangle + \lambda \overline{\lambda} \|v\|^2 \\
 & = \|u\|^2 -  \frac{|\langle u, v \rangle|^2}{\|v\|^2} - \frac{|\langle u, v \rangle|^2}{\|v\|^2} + \frac{|\langle u, v \rangle|^2}{\|v\|^2} \\
& = \|u\|^2 -  \frac{|\langle u, v \rangle|^2}{\|v\|^2}. 
\end{align}
</math>

Therefore, <math>0 \leq \|u\|^2 -  \frac{|\langle u, v \rangle|^2}{\|v\|^2}</math>, or <math>|\langle u, v \rangle| \leq \|u\| \|v\|</math>.

If the inequality holds as an equality, then <math>\| u - \lambda \cdot v \| = 0</math>, and so <math>u - \lambda \cdot v = 0</math>, thus <math>u</math> and <math>v</math> are linearly dependent. On the other hand, if <math>u</math> and <math>v</math> are linearly dependent, then <math>|\langle u, v \rangle| = \|u\| \|v\|</math>, as shown in the first proof.

=== More proofs ===
There are many different proofs<ref>{{cite journal |last1=Wu |first1=Hui-Hua |last2=Wu |first2=Shanhe |date=April 2009 |title=Various proofs of the Cauchy-Schwarz inequality |journal=Octogon Mathematical Magazine |issn=1222-5657 |isbn=978-973-88255-5-0 |volume=17 |issue=1 |pages=221–229 |url=http://www.uni-miskolc.hu/~matsefi/Octogon/volumes/volume1/article1_19.pdf |access-date=18 May 2016}}</ref> of the Cauchy–Schwarz inequality other than the above two examples.<ref name="Steele" /><ref name=":0" /> When consulting other sources, there are often two sources of confusion. First, some authors define {{math|⟨⋅,⋅⟩}} to be linear in the [[Inner product space#Alternative definitions, notations and remarks|second argument]] rather than the first. Second, some proofs are only valid when the field is <math>\mathbb R</math> and not <math>\mathbb C</math>.<ref>{{cite book |last1=Aliprantis |first1=Charalambos D. |last2=Border |first2=Kim C. |date=2007-05-02 |title=Infinite Dimensional Analysis: A Hitchhiker's Guide |publisher=Springer Science & Business Media |isbn=9783540326960 |url=https://books.google.com/books?id=4hIq6ExH7NoC}}</ref>

==Special cases==

===Titu's lemma===
Titu's lemma (named after [[Titu Andreescu]], also known as T2 lemma, Engel's form, or Sedrakyan's inequality) states that for positive reals, one has

:<math> \frac{\left( \sum_{i=1}^n u_i \right)^2 }{\sum_{i=1}^n v_i} \leq \sum_{i=1}^n \frac{u_i^2}{v_i}. </math>

It is a direct consequence of the Cauchy–Schwarz inequality, obtained upon substituting <math>u_i' = \frac{u_i}{\sqrt{v_i}}</math> and <math>v_i' =  \sqrt{v_i}.</math> This form is especially helpful when the inequality involves fractions where the numerator is a perfect square.

===R<sup>2</sup> (ordinary two-dimensional space)===
In the usual 2-dimensional space with the [[dot product]], let <math>v = (v_1,v_2)</math> and <math>u = (u_1,u_2)</math>. The Cauchy–Schwarz inequality is that
:<math>\langle u, v \rangle^2 = (\|u\| \|v\| \cos \theta)^2 \leq \|u\|^2 \|v\|^2,</math>
where <math>\theta</math> is the [[angle]] between <math>u</math> and <math>v.</math>

The form above is perhaps the easiest in which to understand the inequality, since the square of the cosine can be at most 1, which occurs when the vectors are in the same or opposite directions. It can also be restated in terms of the vector coordinates <math>v_1, v_2, u_1</math> and <math>u_2</math> as
: <math>(u_1v_1 + u_2v_2)^2 \leq (u_1^2 + u_2^2) (v_1^2 + v_2^2),</math>
where equality holds if and only if the vector <math>(u_1,u_2)</math> is in the same or opposite direction as the vector <math>(v_1, v_2),</math> or if one of them is the zero vector.

=== R<sup>''n''</sup> (''n''-dimensional Euclidean space)===
In [[Euclidean space]] <math>\mathbb R ^n</math> with the standard inner product, the Cauchy–Schwarz inequality is
: <math>\left(\sum_{i=1}^n u_i v_i\right)^2\leq \left(\sum_{i=1}^n u_i^2\right) \left(\sum_{i=1}^n v_i^2\right)</math>

The Cauchy–Schwarz inequality can be proved using only ideas from elementary algebra in this case. 
Consider the following quadratic polynomial in <math>x</math>
: <math>0 \leq (u_1 x + v_1)^2 + \cdots + (u_n x + v_n)^2 = \left( \sum u_i^2 \right) x^2 + 2 \left( \sum u_i v_i \right) x + \sum v_i^2.</math>
Since it is nonnegative, it has at most one real root for <math>x</math>, hence its [[discriminant]] is less than or equal to zero. That is,
: <math>\left(\sum ( u_i v_i ) \right)^2 - \sum {u_i^2}  \sum {v_i^2} \le 0,</math>
which yields the Cauchy–Schwarz inequality.

===L<sup>2</sup>===
For the inner product space of [[square-integrable]] complex-valued [[function (mathematics)|functions]], one has
: <math>\left|\int_{\mathbb{R}^n} f(x) \overline{g(x)}\,dx\right|^2 \leq \int_{\mathbb{R}^n} |f(x)|^2\,dx  \int_{\mathbb{R}^n} |g(x)|^2 \,dx.</math>
A generalization of this is the [[Hölder inequality]].

== Applications ==

=== Analysis ===
The [[triangle inequality]] for the standard norm is often shown as a consequence of the Cauchy–Schwarz inequality, as follows: given vectors ''x'' and ''y'':
: <math>\begin{align}
\|x + y\|^2 & = \langle x + y, x + y \rangle \\
& = \|x\|^2 + \langle x, y \rangle + \langle y, x \rangle + \|y\|^2 \\
& = \|x\|^2 + 2 \operatorname{Re} \langle x, y \rangle + \|y\|^2\\
& \le \|x\|^2 + 2|\langle x, y \rangle| + \|y\|^2 \\
& \le \|x\|^2 + 2\|x\|\|y\| + \|y\|^2 \\
& = (\|x\| + \|y\|)^2
\end{align}</math>

Taking square roots gives the triangle inequality.

The Cauchy–Schwarz inequality is used to prove that the inner product is a [[continuous function]] with respect to the [[topology]] induced by the inner product itself.<ref>{{cite book |last1=Bachman |first1=George |last2=Narici |first2=Lawrence |date=2012-09-26 |title=Functional Analysis |publisher=Courier Corporation |isbn=9780486136554 |pages=141|url=https://books.google.com/books?id=_lTDAgAAQBAJ}}</ref><ref>{{cite book |last=Swartz |first=Charles |date=1994-02-21 |title=Measure, Integration and Function Spaces |publisher=World Scientific |isbn=9789814502511 |pages=236 |url=https://books.google.com/books?id=SsbsCgAAQBAJ}}</ref>

=== Geometry ===
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:<ref>{{cite book |last=Ricardo |first=Henry |date=2009-10-21 |title=A Modern Introduction to Linear Algebra |publisher=CRC Press |isbn=9781439894613 |pages=18 |url=https://books.google.com/books?id=s7bMBQAAQBAJ}}</ref><ref>{{cite book |last1=Banerjee |first1=Sudipto |last2=Roy |first2=Anindya |date=2014-06-06 |title=Linear Algebra and Matrix Analysis for Statistics |publisher=CRC Press |isbn=9781482248241 |pages=181 |url=https://books.google.com/books?id=WDTcBQAAQBAJ}}</ref>
: <math>\cos\theta_{xy}=\frac{\langle x,y\rangle}{\|x\| \|y\|}.</math>
The Cauchy–Schwarz inequality proves that this definition is sensible, by showing that the right-hand side lies in the interval [&minus;1,&nbsp;1] and justifies the notion that (real) [[Hilbert space]]s are simply generalizations of the [[Euclidean space]]. It can also be used to define an angle in [[complex numbers|complex]] [[inner-product space]]s, by taking the absolute value or the real part of the right-hand side,<ref>{{cite book |last=Valenza |first=Robert J. |date=2012-12-06 |title=Linear Algebra: An Introduction to Abstract Mathematics |publisher=Springer Science & Business Media |isbn=9781461209010 |pages=146 |url=https://books.google.com/books?id=7x8MCAAAQBAJ}}</ref><ref>{{cite book |last=Constantin |first=Adrian |date=2016-05-21 |title=Fourier Analysis with Applications |publisher=Cambridge University Press |isbn=9781107044104 |pages=74 |url=https://books.google.com/books?id=JnMZDAAAQBAJ}}</ref> as is done when extracting a metric from [[Fidelity of quantum states|quantum fidelity]].

===Probability theory===
<!-- For the multivariate case,{{clarify|reason=define GE operator here|date=July 2011}}<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|doi=10.1016/s0165-1765(99)00014-2}}</ref>
: <math>\operatorname{Var}(Y) \ge \operatorname{Cov} (Y,X) \operatorname{Var}^{-1}(X) \operatorname{Cov}(X,Y)</math>
This inequality means that the difference is semidefinite positive. -->

Let ''X'', ''Y'' be [[random variable]]s, then the covariance inequality<ref>{{cite book |last=Mukhopadhyay |first=Nitis |date=2000-03-22 |title=Probability and Statistical Inference |publisher=CRC Press |isbn=9780824703790 |pages=150 |url=https://books.google.com/books?id=TMSnGkr_DxwC}}</ref><ref>{{cite book |last=Keener |first=Robert W. |date=2010-09-08 |title=Theoretical Statistics: Topics for a Core Course |publisher=Springer Science & Business Media |isbn=9780387938394 |pages=71 |url=https://books.google.com/books?id=aVJmcega44cC}}</ref> is given by
: <math>\operatorname{Var}(Y)\ge\frac{\operatorname{Cov}(Y,X)\operatorname{Cov}(Y,X)}{\operatorname{Var}(X)}.</math>
After defining an inner product on the set of random variables using the expectation of their product,
: <math>\langle X, Y \rangle := \operatorname{E}(X Y),</math>
the Cauchy–Schwarz inequality becomes
: <math>|\operatorname{E}(XY)|^2 \leq \operatorname{E}(X^2) \operatorname{E}(Y^2).</math>
To prove the covariance inequality using the Cauchy–Schwarz inequality, let <math>\mu = \operatorname{E}(X)</math> and <math>\nu = \operatorname{E}(Y)</math>, then
: <math>\begin{align}
|\operatorname{Cov}(X,Y)|^2 &= |\operatorname{E}( (X - \mu)(Y - \nu) )|^2 \\
&= | \langle X - \mu, Y - \nu \rangle |^2\\
&\leq \langle X - \mu, X - \mu \rangle \langle Y - \nu, Y - \nu \rangle \\
& = \operatorname{E}( (X-\mu)^2 ) \operatorname{E}( (Y-\nu)^2 ) \\
& = \operatorname{Var}(X) \operatorname{Var}(Y),
\end{align}</math>

where <math>\operatorname{Var}</math> denotes [[variance]], and <math>\operatorname{Cov}</math> 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 are replaced by a [[C*-algebra]] or [[W*-algebra]].

An inner product can be used to define a [[positive linear functional]]. For example, given a Hilbert space <math>L^2(m), m</math> being a finite measure, the standard inner product gives rise to a positive functional <math>\varphi</math> by <math>\varphi (g) = \langle g, 1 \rangle</math>.  Conversely, every positive linear functional <math>\varphi</math> on <math>L^2(m)</math> can be used to define an  inner product <math>\langle f, g \rangle _\varphi := \varphi(g^*f)</math>, where <math>g^*</math> is the [[Pointwise product|pointwise]] [[complex conjugate]] of <math>g</math>. In this language, the Cauchy–Schwarz inequality becomes<ref>{{cite book |last1=Faria |first1=Edson de |last2=Melo |first2=Welington de |date=2010-08-12 |title=Mathematical Aspects of Quantum Field Theory |publisher=Cambridge University Press |isbn=9781139489805 |pages=273 |url=https://books.google.com/books?id=u9M9PFLNpMMC}}</ref>
: <math>| \varphi(g^*f) |^2 \leq \varphi(f^*f) \varphi(g^*g),</math>
which extends verbatim to positive functionals on C*-algebras:

'''Theorem''' (Cauchy–Schwarz inequality for  positive functionals on C*-algebras):<ref>{{cite book |last=Lin |first=Huaxin |date=2001-01-01 |title=An Introduction to the Classification of Amenable C*-algebras |publisher=World Scientific |isbn=9789812799883 |pages=27 |url=https://books.google.com/books?id=2qru8d7BCAAC}}</ref><ref>{{cite book |last=Arveson |first=W. |date=2012-12-06 |title=An Invitation to C*-Algebras |publisher=Springer Science & Business Media |isbn=9781461263715 |pages=28 |url=https://books.google.com/books?id=d5TqBwAAQBAJ}}</ref> If <math>\varphi</math> is a positive linear functional on a C*-algebra <math>A,</math> then for all <math>a, b \in A</math>, <math>|\varphi(b^*a)|^2 \leq \varphi(b^*b)\varphi(a^*a)</math>.

The next two theorems are further examples in operator algebra.

'''Theorem''' (Kadison–Schwarz inequality,<ref>{{cite book |last=Størmer |first=Erling |date=2012-12-13 |title=Positive Linear Maps of Operator Algebras |series=Springer Monographs in Mathematics |publisher=Springer Science & Business Media |isbn=9783642343698 |url=https://books.google.com/books?id=lQtKAIONqwIC}}</ref><ref>{{cite journal |last=Kadison |first=Richard V. |date=1952-01-01 |title=A Generalized Schwarz Inequality and Algebraic Invariants for Operator Algebras |jstor=1969657 |journal=Annals of Mathematics |doi=10.2307/1969657 |volume=56 |number=3 |pages=494–503}}</ref> named after [[Richard Kadison]]): If <math>\varphi</math> is a unital positive map, then for every [[normal operator|normal element]] <math>a</math> in its domain, we have <math>\varphi(a^*a) \ge \varphi(a^*) \varphi(a)</math> and <math>\varphi(a^*a) \ge \varphi(a) \varphi(a^*)</math>.

This extends the fact <math>\varphi(a^*a) \cdot 1 \ge \varphi(a)^* \varphi(a) = |\varphi(a)|^2</math>, when <math>\varphi</math> is a linear functional. The case when <math>a</math> is self-adjoint, i.e. <math>a = a^*,</math> is sometimes known as '''Kadison's inequality'''.

'''Theorem''' (Modified Schwarz inequality for 2-positive maps):<ref>{{cite book |last=Paulsen |first=Vern |year=2002 |title=Completely Bounded Maps and Operator Algebras |series=Cambridge Studies in Advanced Mathematics |volume=78 |publisher=Cambridge University Press |isbn=9780521816694 |page=40 |url=https://books.google.com/books?id=VtSFHDABxMIC&pg=PA40}}</ref> For a 2-positive map <math>\varphi</math> between C*-algebras, for all <math>a, b</math> in its domain,
: <math>\varphi(a)^*\varphi(a) \leq \Vert\varphi(1)\Vert\varphi(a^*a), \text{ and }</math>
: <math>\Vert\varphi(a^*b)\Vert^2 \leq \Vert\varphi(a^*a)\Vert \cdot \Vert\varphi(b^*b)\Vert.</math>
 
Another generalization is a refinement obtained by interpolating between both sides the Cauchy-Schwarz inequality:  

'''Theorem''' (Callebaut's Inequality)<ref>{{cite journal |last1=Callebaut|first1=D.K. |date=1965 |title=Generalization of the Cauchy–Schwarz inequality   |journal=J. Math. Anal. Appl. |volume=12  |issue=3 |pages=491–494 |doi=10.1016/0022-247X(65)90016-8}}</ref> 
For reals <math>0\leqslant s\leqslant t\leqslant 1</math>, 
: <math>\Bigl(\sum_{i=1}^{n}a_{i}b_{i} \Bigr) ^2\leqslant\sum_{i=1}^{n}a_{i}^{1+s}b_{i}^{1-s}\sum_{i=1}^{n}a_{i}^{1-s}b_{i}^{1+s} \leqslant\sum_{i=1}^{n}a_{i}^{1+t}b_{i}^{1-t}\sum_{i=1}^{n}a_{i}^{1-t}b_{i}^{1+t} \leqslant \sum_{i=1}^{n}a_{i}^2\sum_{i=1}^{n}b_{i}^2 . </math>

It can be easily proven by [[Hölder's inequality]].<ref>{{cite book  |title=Callebaut's inequality |publisher=Entry in the AoPS Wiki |url=https://artofproblemsolving.com/wiki/index.php?title=Callebaut%27s_Inequality}}</ref> There are also non commutative versions for operators and tensor products of matrices.<ref>{{cite arxiv|last1= Moslehian |first1=M.S. |last2=Matharu |first2=J.S. |last3=Aujla |first3=J.S. |date=2011 |title=Non-commutative Callebaut inequality |eprint=1112.3003 |class=math.FA }}</ref>

== See also ==
* [[Bessel's inequality]]
* [[Jensen's inequality]]
* [[Kunita–Watanabe inequality]]
* [[Minkowski inequality]]

==Notes==
{{Reflist|30em}}

==References==
{{refbegin}}
* {{citation|first=J. M.|last=Aldaz|first2=S.|last2=Barza|first3=M.|last3=Fujii|first4=M. S.|last4=Moslehian|title=Advances in Operator Cauchy—Schwarz inequalities and their reverses|journal=Annals of Functional Analysis|volume=6|year=2015|issue=3|pages=275–295|doi=10.15352/afa/06-3-20}}
* {{springer|id=b/b017770|title=Bunyakovskii inequality|first=V. I.|last=Bityutskov}}
* {{citation|first=V.|last=Bunyakovsky |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}}
* {{citation|first=A.-L.|last=Cauchy|title= Sur les formules qui résultent de l'emploie du signe et sur > ou <, et sur les moyennes entre plusieurs quantités |journal= Cours d'Analyse, 1er Partie: Analyse Algébrique 1821; OEuvres Ser.2 III 373-377|year=1821}}
* {{citation|first=S. S.|last=Dragomir|title=A survey on Cauchy–Bunyakovsky–Schwarz type discrete inequalities|journal=Journal of Inequalities in Pure and Applied Mathematics|volume=4|issue=3|year=2003|pages=142 pp|url=http://jipam.vu.edu.au/article.php?sid=301|url-status=dead|archiveurl=https://web.archive.org/web/20080720034744/http://jipam.vu.edu.au/article.php?sid=301|archivedate=2008-07-20}}
*{{Citation | last1=Grinshpan | first1=A. Z. | title=General inequalities, consequences, and applications | doi=10.1016/j.aam.2004.05.001 | year=2005 | 
journal=Advances in Applied Mathematics | volume=34 | issue=1 | pages=71–100 }}
* {{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}}
* {{springer|title=Cauchy inequality|id=C/c020880|first=E. D.|last=Solomentsev}}
* {{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}}
{{refend}}

==External links==
* [http://jeff560.tripod.com/c.html Earliest Uses: The entry on the Cauchy–Schwarz inequality has some historical information.]
* [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.

{{Functional Analysis}}

{{DEFAULTSORT:Cauchy-Schwarz inequality}}
[[Category:Augustin-Louis Cauchy]]
[[Category:Inequalities]]
[[Category:Linear algebra]]
[[Category:Operator theory]]
[[Category:Articles containing proofs]]
[[Category:Mathematical analysis]]
[[Category:Probabilistic inequalities]]