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{{short description|Mathematical inequality relating inner products and norms}}
The '''Cauchy–Schwarz inequality''' (also called '''Cauchy–Bunyakovsky–Schwarz inequality''')<ref>{{cite web|url=https://mathshistory.st-andrews.ac.uk/Biographies/Schwarz/|title=Hermann Amandus Schwarz|first1=J.J.|last1=O'Connor|first2=E.F.|last2=Robertson|website=[[University of St Andrews]], Scotland }}</ref><ref name="EncyclopediaOfMath">{{springer|id=b/b017770|title=Bunyakovskii inequality|first=V. I.|last=Bityutskov}}</ref><ref>{{cite web|url=http://www.wwu.edu/faculty/curgus/Courses/Math_pages/Math_504/Cauchy-Schwarz-Bunyakovsky.html|title=Cauchy-Bunyakovsky-Schwarz inequality|first=Branko|last=Ćurgus|website=Western Washington University|department=Department of Mathematics }}</ref><ref>{{cite web|url=https://mathcs.clarku.edu/~djoyce/ma130/cauchy.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://mathcs.clarku.edu/~djoyce/ma130/cauchy.pdf |archive-date=2022-10-09 |url-status=live|title=Cauchy's inequality|first=David E.|last=Joyce|website=Clark University|department=Department of Mathematics and Computer Science }}</ref> is an upper bound on the [[Inner product space|inner product]] between two [[Vector (mathematics and physics)|vectors]] in an inner product space in terms of the product of the vector [[Norm (mathematics)|norms]]. It is considered one of the most important and widely used [[Inequality (mathematics)|inequalities]] in 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>

Inner products of vectors can describe [[finite sum]]s (via finite-dimensional vector spaces), [[infinite series]] (via vectors in [[sequence space]]s), and [[integral]]s (via vectors in [[Hilbert space]]s). The inequality for sums was published by {{harvs|first=Augustin-Louis|last=Cauchy|authorlink=Augustin-Louis Cauchy|year=1821|txt=yes}}. The corresponding inequality for integrals was published by {{harvs|first=Viktor|last=Bunyakovsky|author-link=Viktor Yakovlevich Bunyakovsky|txt=yes|year=1859}}<ref name="EncyclopediaOfMath"/> and {{harvs|txt=yes|authorlink=Hermann Schwarz|first=Hermann|last=Schwarz|year=1888}}. Schwarz gave the modern proof of the integral version.<ref name="Steele" />

== Statement of the inequality ==
The Cauchy–Schwarz inequality states that for all vectors <math>\mathbf{u}</math> and <math>\mathbf{v}</math> of an [[inner product space]]

{{NumBlk|:|<math>\left |\langle \mathbf{u}, \mathbf{v} \rangle\right |^2 \leq \langle \mathbf{u}, \mathbf{u} \rangle \cdot \langle \mathbf{v}, \mathbf{v} \rangle,</math>|{{EquationRef|Cauchy–Schwarz inequality [written using only the inner product]|1}}}}
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]]. Every inner product gives rise to a Euclidean <math>\ell_2</math> [[Norm (mathematics)|norm]], called the {{em|canonical}} or [[inner product space#Norm|{{em|induced}} {{em|norm}}]], where the norm of a vector <math>\mathbf{u}</math> is denoted and defined by
<math display="block">\|\mathbf{u}\| := \sqrt{\langle \mathbf{u}, \mathbf{u} \rangle},</math> where <math>\langle \mathbf{u}, \mathbf{u} \rangle</math> is always a non-negative real number (even if the inner product is complex-valued). 
By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality can be written in its more familiar form in terms of the norm:<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>

{{NumBlk|:|<math>|\langle \mathbf{u}, \mathbf{v} \rangle| \leq \|\mathbf{u}\|  \|\mathbf{v}\|.</math>|{{EquationRef|Cauchy–Schwarz inequality - written using norm and inner product|2}}}}

Moreover, the two sides are equal if and only if <math>\mathbf{u}</math> and <math>\mathbf{v}</math> are [[linear independence|linearly dependent]].<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{{pipe}}c>&nbsp;=&nbsp;0 or {{pipe}}c>&nbsp;=&nbsp;0. From the definition of {{pipe}}c>, we conclude that {{pipe}}a> and {{pipe}}b> must be proportional.}}</ref><ref>{{cite book|last1=Axler|first1=Sheldon|date=2015|title=Linear Algebra Done Right, 3rd Ed.|publisher=Springer International Publishing|isbn=978-3-319-11079-0|page=172|url=https://books.google.com/books?id=CQWwoQEACAAJ|quote=This inequality is an equality if and only if one of ''u'', ''v'' is a scalar multiple of the other.}}</ref>

== Special cases ==

=== Sedrakyan's lemma – positive real numbers ===
[[Sedrakyan's inequality]], also known as [[Harald Bergström|Bergström]]'s inequality, [[Arthur Engel (mathematician)|Engel]]'s form, [[Titu Andreescu|Titu]]'s lemma (or the T2 lemma), states that for real numbers <math>u_1, u_2, \dots, u_n</math> and positive real numbers <math>v_1, v_2, \dots, v_n</math>:
<math display=block>\frac{\left(u_1 + u_2 + \cdots + u_n\right)^2}{v_1 + v_2 + \cdots + v_n} \leq \frac{u^2_1}{v_1} + \frac{u^2_2}{v_2} + \cdots + \frac{u^2_n}{v_n},</math>
or, using summation notation,
<math display=block>\biggl(\sum_{i=1}^n u_i\biggr)^2 \bigg/ \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 by using the [[dot product]] on <math>\R^n</math> upon substituting <math>u_i' = \frac{u_i}{\sqrt{v_i\vphantom{t}}}</math> and <math>v_i' = {\textstyle \sqrt{v_i\vphantom{t}}}</math>. This form is especially helpful when the inequality involves fractions where the numerator is a [[Square number|perfect square]].

=== {{math|R<sup>2</sup>}} - The plane ===
[[File:Cauchy-Schwarz inequation in Euclidean plane.gif|thumb|300px|Cauchy–Schwarz inequality in a unit circle of the Euclidean plane]]
The real vector space <math>\R^2</math> denotes the 2-dimensional plane. It is also the 2-dimensional [[Euclidean space]] where the inner product is the [[dot product]]. 
If <math>\mathbf{u} = (u_1, u_2)</math> and <math>\mathbf{v} = (v_1, v_2)</math> then the Cauchy–Schwarz inequality becomes:
<math display=block>\langle \mathbf{u}, \mathbf{v} \rangle^2 = \bigl(\|\mathbf{u}\| \|\mathbf{v}\| \cos \theta\bigr)^2 \leq \|\mathbf{u}\|^2 \|\mathbf{v}\|^2,</math>
where <math>\theta</math> is the [[angle]] between <math>\mathbf{u}</math> and <math>\mathbf{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>u_1</math>, <math>u_2</math>, <math>v_1</math>, and <math>v_2</math> as
<math display=block>\left(u_1 v_1 + u_2 v_2\right)^2 \leq \left(u_1^2 + u_2^2\right) \left(v_1^2 + v_2^2\right),</math>
where equality holds if and only if the vector <math>\left(u_1, u_2\right)</math> is in the same or opposite direction as the vector <math>\left(v_1, v_2\right)</math>, or if one of them is the zero vector.

=== {{anchor|Real Euclidean space}}{{math|'''R'''<sup>''n''</sup>}}: ''n''-dimensional Euclidean space ===
{{anchor|real number proof}}In [[Euclidean space]] <math>\R^n</math> with the standard inner product, which is the [[dot product]], the Cauchy–Schwarz inequality becomes:
<math display=block>\biggl(\sum_{i=1}^n u_i v_i\biggr)^2 \leq \biggl(\sum_{i=1}^n u_i^2\biggr) \biggl(\sum_{i=1}^n v_i^2\biggr).</math>

The Cauchy–Schwarz inequality can be proved using only elementary algebra in this case by observing that the difference of the right and the left hand side is 
<math display=block> \tfrac{1}{2} \sum_{i=1}^n\sum_{j=1}^n (u_i v_j - u_j v_i)^2 \ge 0</math> 

or by considering the following [[quadratic polynomial]] in <math>x</math>
<math display=block> (u_1 x + v_1)^2 + \cdots + (u_n x + v_n)^2 = \biggl(\sum_i u_i^2\biggr) x^2 + 2 \biggl(\sum_i u_i v_i\biggr) x + \sum_i v_i^2.</math>

Since the latter polynomial is nonnegative, it has at most one real root, hence its [[discriminant]] is less than or equal to zero. That is,
<math display=block>\biggl(\sum_i u_i v_i\biggr)^2 - \biggl(\sum_i {u_i^2}\biggr)  \biggl(\sum_i {v_i^2}\biggr) \leq 0.</math>

=== {{math|'''C'''<sup>''n''</sup>}}: ''n''-dimensional complex space===
If <math>\mathbf{u}, \mathbf{v} \in \Complex^n</math> with <math>\mathbf{u} = (u_1, \ldots, u_n)</math> and <math>\mathbf{v} = (v_1, \ldots, v_n)</math> (where <math>u_1, \ldots, u_n \in \Complex</math> and <math>v_1, \ldots, v_n \in \Complex</math>) and if the inner product on the vector space <math>\Complex^n</math> is the canonical complex inner product (defined by <math>\langle \mathbf{u}, \mathbf{v} \rangle := u_1 \overline{v_1} + \cdots + u_{n} \overline{v_n},</math> where the bar notation is used for [[Complex conjugate|complex conjugation]]), then the inequality may be restated more explicitly as follows: 
<math display=block>\bigl|\langle \mathbf{u}, \mathbf{v} \rangle\bigr|^2 
= \Biggl|\sum_{k=1}^n u_k\bar{v}_k\Biggr|^2 
\leq \langle \mathbf{u}, \mathbf{u} \rangle \langle \mathbf{v}, \mathbf{v} \rangle 
= \biggl(\sum_{k=1}^n u_k \bar{u}_k\biggr) \biggl(\sum_{k=1}^n v_k \bar{v}_k\biggr) 
= \sum_{j=1}^n |u_j|^2 \sum_{k=1}^n |v_k|^2.</math>

That is,
<math display=block>\bigl|u_1 \bar{v}_1 + \cdots + u_n \bar{v}_n\bigr|^2 \leq \bigl(|u_1|{}^2 + \cdots + |u_n|{}^2\bigr) \bigl(|v_1|{}^2 + \cdots + |v_n|{}^2\bigr).</math>

=== {{math|''L''<sup>2</sup>}} ===

For the inner product space of [[square-integrable]] complex-valued [[function (mathematics)|functions]], the following inequality holds.
<math display=block>\left|\int_{\R^n} f(x) \overline{g(x)}\,dx\right|^2  \leq  \int_{\R^n} \bigl|f(x)\bigr|^2\,dx \int_{\R^n} \bigl|g(x)\bigr|^2 \,dx.</math>

The [[Hölder inequality]] is a generalization of this.

== Applications ==

=== Analysis ===
In any [[inner product space]], the [[triangle inequality]] is a consequence of the Cauchy–Schwarz inequality, as is now shown: 
<math display="block">\begin{alignat}{4}
\|\mathbf{u} + \mathbf{v}\|^2 
&= \langle \mathbf{u} + \mathbf{v}, \mathbf{u} + \mathbf{v} \rangle && \\
&= \|\mathbf{u}\|^2 + \langle \mathbf{u}, \mathbf{v} \rangle + \langle \mathbf{v}, \mathbf{u} \rangle + \|\mathbf{v}\|^2 ~ && ~ \text{ where } \langle \mathbf{v}, \mathbf{u} \rangle = \overline{\langle \mathbf{u}, \mathbf{v} \rangle} \\
&= \|\mathbf{u}\|^2 + 2 \operatorname{Re} \langle \mathbf{u}, \mathbf{v} \rangle + \|\mathbf{v}\|^2 && \\
&\leq \|\mathbf{u}\|^2 + 2|\langle \mathbf{u}, \mathbf{v} \rangle| + \|\mathbf{v}\|^2 && \\
&\leq \|\mathbf{u}\|^2 + 2\|\mathbf{u}\|\|\mathbf{v}\| + \|\mathbf{v}\|^2 ~ && ~ \text{ using CS}\\
&=\bigl(\|\mathbf{u}\| + \|\mathbf{v}\|\bigr)^2. && 
\end{alignat}</math>

Taking square roots gives the triangle inequality:
<math display=block>\|\mathbf{u} + \mathbf{v}\| \leq \|\mathbf{u}\| + \|\mathbf{v}\|.</math>

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 display=block>\cos\theta_{\mathbf{u} \mathbf{v}} = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\|\mathbf{u}\| \|\mathbf{v}\|}.</math>

The Cauchy–Schwarz inequality proves that this definition is sensible, by showing that the right-hand side lies in the interval {{math|[&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 |archive-url=https://ghostarchive.org/archive/20221009/http://web2.uconn.edu/tripathi/published-papers/cs.pdf |archive-date=2022-10-09 |url-status=live|doi=10.1016/s0165-1765(99)00014-2}}</ref>
<math display=block>\operatorname{Var}(Y) \geq \operatorname{Cov} (Y, X) \operatorname{Var}^{-1}(X) \operatorname{Cov}(X, Y)</math>
This inequality means that the difference is semidefinite positive. -->

Let <math>X</math> and <math>Y</math> 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 display=block>\operatorname{Var}(X) \geq \frac{\operatorname{Cov}(X, Y)^2}{\operatorname{Var}(Y)}.</math>

After defining an inner product on the set of random variables using the expectation of their product,
<math display=block>\langle X, Y \rangle := \operatorname{E}(X Y),</math>
the Cauchy–Schwarz inequality becomes
<math display=block>\bigl|\operatorname{E}(XY)\bigr|^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 display=block>\begin{align}
\bigl|\operatorname{Cov}(X, Y)\bigr|^2 
&= \bigl|\operatorname{E}((X - \mu)(Y - \nu))\bigr|^2 \\
&= \bigl|\langle X - \mu, Y - \nu \rangle \bigr|^2\\
&\leq \langle X - \mu, X - \mu \rangle \langle Y - \nu, Y - \nu \rangle \\
& = \operatorname{E}\left((X - \mu)^2\right) \operatorname{E}\left((Y - \nu)^2\right) \\
& = \operatorname{Var}(X) \operatorname{Var}(Y),
\end{align}</math>
where <math>\operatorname{Var}</math> denotes [[variance]] and <math>\operatorname{Cov}</math> denotes [[covariance]].

==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 |archive-url=https://ghostarchive.org/archive/20221009/http://www.uni-miskolc.hu/~matsefi/Octogon/volumes/volume1/article1_19.pdf |archive-date=2022-10-09 |url-status=live|access-date=18 May 2016}}</ref> of the Cauchy–Schwarz inequality other than those given below.<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>

This section gives two proofs of the following theorem:

{{math theorem|name=Cauchy–Schwarz inequality|note=|style=|math_statement= 
Let <math>\mathbf{u}</math> and <math>\mathbf{v}</math> be arbitrary vectors in an [[inner product space]] over the scalar field <math>\mathbb{F},</math> where <math>\mathbb F</math> is the field of real numbers <math>\R</math> or complex numbers <math>\Complex.</math> Then

{{NumBlk|:|<math>\bigl|\langle \mathbf{u}, \mathbf{v} \rangle\bigr| \leq \|\mathbf{u}\| \|\mathbf{v}\|</math>|{{EquationRef|Cauchy–Schwarz Inequality}}}}

with {{visible anchor|Characterization of Equality in Cauchy–Schwarz|text=equality holding}} in the {{EquationNote|Cauchy–Schwarz Inequality}} [[if and only if]] <math>\mathbf{u}</math> and <math>\mathbf{v}</math> are [[Linear independence|linearly dependent]]. 

Moreover, if <math> \left| \langle \mathbf{u}, \mathbf{v} \rangle\right| = \|\mathbf{u}\| \|\mathbf{v}\|</math> and <math>\mathbf{v} \neq \mathbf{0}</math> then <math>\mathbf{u} = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\|\mathbf{v}\|^2} \mathbf{v}.</math>
}} 


In both of the proofs given below, the proof in the trivial case where at least one of the vectors is zero (or equivalently, in the case where <math>\|\mathbf{u}\|\|\mathbf{v}\|= 0</math>) is the same. It is presented immediately below only once to reduce repetition. It also includes the easy part of the proof of the [[#Characterization of Equality in Cauchy–Schwarz|Equality Characterization]] given above; that is, it proves that if <math>\mathbf{u}</math> and <math>\mathbf{v}</math> are linearly dependent then <math>\bigl|\langle \mathbf{u}, \mathbf{v} \rangle\bigr| = \|\mathbf{u}\| \|\mathbf{v}\|.</math> 

{{collapse top|title=Proof of the trivial parts: Case where a vector is <math>\mathbf{0}</math> and also one direction of the [[#Characterization of Equality in Cauchy–Schwarz|Equality Characterization]]|left=true}}
<!-- Note to Editors:
Anyone reading this part of the proof is very likely to be new to this subject, which is why this proof is much more detailed than the proofs given elsewhere in this article.
-->
By definition, <math>\mathbf{u}</math> and <math>\mathbf{v}</math> are linearly dependent if and only if one is a scalar multiple of the other. 
If <math>\mathbf{u} = c \mathbf{v}</math> where <math>c</math> is some scalar then 
<math display=block>|\langle \mathbf{u}, \mathbf{v} \rangle| 
= |\langle c \mathbf{v}, \mathbf{v} \rangle| 
= |c \langle \mathbf{v}, \mathbf{v} \rangle| 
= |c|\|\mathbf{v}\| \|\mathbf{v}\|
=\|c \mathbf{v}\| \|\mathbf{v}\|
=\|\mathbf{u}\| \|\mathbf{v}\|</math>

which shows that equality holds in the {{EquationNote|Cauchy–Schwarz Inequality}}. 
The case where <math>\mathbf{v} = c \mathbf{u}</math> for some scalar <math>c</math> follows from the previous case:
<math display=block>|\langle \mathbf{u}, \mathbf{v} \rangle| 
= |\langle \mathbf{v}, \mathbf{u} \rangle|
=\|\mathbf{v}\| \|\mathbf{u}\|.</math>

In particular, if at least one of <math>\mathbf{u}</math> and <math>\mathbf{v}</math> is the zero vector then <math>\mathbf{u}</math> and <math>\mathbf{v}</math> are necessarily linearly dependent (for example, if <math>\mathbf{u} = \mathbf{0}</math> then <math>\mathbf{u} = c \mathbf{v}</math> where <math>c = 0</math>), so the above computation shows that the Cauchy–Schwarz inequality holds in this case. 
{{collapse bottom}}

Consequently, the Cauchy–Schwarz inequality only needs to be proven only for non-zero vectors and also only the non-trivial direction of the [[#Characterization of Equality in Cauchy–Schwarz|Equality Characterization]] must be shown. 

===Proof via the Pythagorean theorem===

The special case of <math>\mathbf{v} = \mathbf{0}</math> was proven above so it is henceforth assumed that <math>\mathbf{v} \neq \mathbf{0}.</math> 
Let
<math display=block>\mathbf{z} := \mathbf{u} - \frac {\langle \mathbf{u}, \mathbf{v} \rangle} {\langle \mathbf{v}, \mathbf{v} \rangle} \mathbf{v}.</math>

It follows from the linearity of the inner product in its first argument that:
<math display=block>\langle \mathbf{z}, \mathbf{v} \rangle 
= \left\langle \mathbf{u} - \frac{\langle \mathbf{u}, \mathbf{v} \rangle} {\langle \mathbf{v}, \mathbf{v} \rangle} \mathbf{v}, \mathbf{v} \right\rangle 
= \langle \mathbf{u}, \mathbf{v} \rangle - \frac{\langle \mathbf{u}, \mathbf{v} \rangle} {\langle \mathbf{v}, \mathbf{v} \rangle} \langle \mathbf{v}, \mathbf{v} \rangle 
= 0.</math>

Therefore, <math>\mathbf{z}</math> is a vector orthogonal to the vector <math>\mathbf{v}</math> (Indeed, <math>\mathbf{z}</math> is the [[vector projection|projection]] of <math>\mathbf{u}</math> onto the plane orthogonal to <math>\mathbf{v}.</math>) We can thus apply the [[Pythagorean theorem#Inner product spaces|Pythagorean theorem]] to
<math display=block>\mathbf{u}= \frac{\langle \mathbf{u}, \mathbf{v} \rangle} {\langle \mathbf{v}, \mathbf{v} \rangle} \mathbf{v} + \mathbf{z}</math>
which gives
<math display=block>\|\mathbf{u}\|^2 
= \left|\frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\langle \mathbf{v}, \mathbf{v} \rangle}\right|^2 \|\mathbf{v}\|^2 + \|\mathbf{z}\|^2 
= \frac{|\langle \mathbf{u}, \mathbf{v} \rangle|^2}{(\|\mathbf{v}\|^2 )^2} \,\|\mathbf{v}\|^2 + \|\mathbf{z}\|^2 
= \frac{|\langle \mathbf{u}, \mathbf{v} \rangle|^2}{\|\mathbf{v}\|^2} + \|\mathbf{z}\|^2 \geq \frac{|\langle \mathbf{u}, \mathbf{v} \rangle|^2}{\|\mathbf{v}\|^2}.</math>

The Cauchy–Schwarz inequality follows by multiplying by <math>\|\mathbf{v}\|^2</math> and then taking the square root. 
Moreover, if the relation <math>\geq</math> in the above expression is actually an equality, then <math>\|\mathbf{z}\|^2 = 0</math> and hence <math>\mathbf{z} = \mathbf{0};</math> the definition of <math>\mathbf{z}</math> then establishes a relation of linear dependence between <math>\mathbf{u}</math> and <math>\mathbf{v}.</math> The converse was proved at the beginning of this section, so the proof is complete. <math>\blacksquare</math>

===Proof by analyzing a quadratic===

Consider an arbitrary pair of vectors <math>\mathbf{u}, \mathbf{v}</math>.  Define the function <math>p : \R \to \R</math> defined by <math>p(t) = \langle t\alpha\mathbf{u} + \mathbf{v}, t\alpha\mathbf{u} + \mathbf{v}\rangle</math>, where <math>\alpha</math> is a complex number satisfying <math>|\alpha| = 1</math> and <math>\alpha\langle\mathbf{u}, \mathbf{v}\rangle = |\langle\mathbf{u}, \mathbf{v}\rangle|</math>.
Such an <math>\alpha</math> exists since if <math>\langle\mathbf{u}, \mathbf{v}\rangle = 0</math> then <math>\alpha</math> can be taken to be 1.

Since the inner product is positive-definite, <math>p(t)</math> only takes non-negative real values. On the other hand, <math>p(t)</math> can be expanded using the bilinearity of the inner product:
<math display=block>\begin{align}
p(t)
&= \langle t\alpha\mathbf{u}, t\alpha\mathbf{u}\rangle + \langle t\alpha\mathbf{u}, \mathbf{v}\rangle + \langle\mathbf{v}, t\alpha\mathbf{u}\rangle + \langle\mathbf{v}, \mathbf{v}\rangle \\
&= t\alpha t\overline{\alpha}\langle\mathbf{u}, \mathbf{u}\rangle + t\alpha\langle\mathbf{u}, \mathbf{v}\rangle + t\overline{\alpha}\langle \mathbf{v}, \mathbf{u}\rangle + \langle\mathbf{v}, \mathbf{v}\rangle \\
&= \lVert \mathbf{u} \rVert^2 t^2 + 2|\langle\mathbf{u}, \mathbf{v}\rangle|t + \lVert \mathbf{v} \rVert^2
\end{align}</math>
Thus, <math>p</math> is a polynomial of degree <math>2</math> (unless <math>\mathbf{u} = 0,</math> which is a case that was checked earlier). Since the sign of <math>p</math> does not change, the discriminant of this polynomial must be non-positive:
<math display=block>\Delta = 4 \bigl(\,|\langle \mathbf{u}, \mathbf{v} \rangle|^2 - \Vert \mathbf{u} \Vert^2 \Vert \mathbf{v} \Vert^2\bigr) \leq 0.</math>
The conclusion follows.<ref>{{Cite book|title=Real and Complex Analysis|last=Rudin|first=Walter|publisher=McGraw-Hill|year=1987|isbn=0070542341|edition=3rd|location=New York|orig-year=1966}}</ref>

For the equality case, notice that <math>\Delta = 0</math> happens if and only if <math>p(t) = \bigl(t\Vert \mathbf{u} \Vert + \Vert \mathbf{v} \Vert\bigr)^2.</math> If <math>t_0 = -\Vert \mathbf{v} \Vert / \Vert \mathbf{u} \Vert,</math> then <math>p(t_0) = \langle t_0\alpha\mathbf{u} + \mathbf{v},t_0\alpha\mathbf{u} + \mathbf{v}\rangle = 0,</math> and hence <math>\mathbf{v} = -t_0\alpha\mathbf{u}.</math>

== Generalizations ==
Various generalizations of the Cauchy–Schwarz inequality exist. [[Hölder's inequality]] generalizes it to <math>L^p</math> norms. More generally, it can be interpreted as a special case of the definition of the norm of a linear operator on a [[Banach space]] (Namely, when the space is a [[Hilbert space]]).  Further generalizations are 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\left(g^* f\right),</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 display=block>\bigl|\varphi(g^* f)\bigr|^2 \leq \varphi\left(f^* f\right) \varphi\left(g^* g\right),</math>

which extends verbatim to positive functionals on C*-algebras:

{{math theorem|name=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>|note=|style=|math_statement= 
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>\left|\varphi\left(b^*a\right)\right|^2 \leq \varphi\left(b^*b\right) \varphi\left(a^*a\right).</math>
}}

The next two theorems are further examples in operator algebra.

{{math theorem|name=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>|note=Named after [[Richard Kadison]]|style=|math_statement= 
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) \geq \varphi\left(a^*\right) \varphi(a)</math> and <math>\varphi\left(a^*a\right) \geq \varphi(a) \varphi\left(a^*\right).</math>
}}

This extends the fact <math>\varphi\left(a^*a\right) \cdot 1 \geq \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, that is, <math>a = a^*,</math> is sometimes known as '''Kadison's inequality'''.

{{math theorem|name=Cauchy–Schwarz inequality|note=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>|style=|math_statement= 
For a 2-positive map <math>\varphi</math> between C*-algebras, for all <math>a, b</math> in its domain,
<math display=block>\begin{align}
\varphi(a)^*\varphi(a) &\leq \Vert\varphi(1)\Vert \varphi\left(a^*a\right), \text{ and } \\[5mu]
\Vert\varphi\left(a^* b\right)\Vert^2 &\leq \Vert\varphi\left(a^*a\right)\Vert \cdot \Vert\varphi\left(b^*b\right)\Vert.
\end{align}</math>
}}

Another generalization is a refinement obtained by interpolating between both sides of the Cauchy–Schwarz inequality:  

{{math theorem|name=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|doi-access=free}}</ref>|note=|style=|math_statement=  
For reals <math>0 \leq s \leq t \leq 1,</math> 
<math display=block>\begin{align}
\biggl(\sum_{i=1}^n a_i b_i\biggr)^2 
~&\leq~ \biggl(\sum_{i=1}^n a_i^{1+s} b_i^{1-s}\biggr) \biggl(\sum_{i=1}^n a_i^{1-s} b_i^{1+s}\biggr) \\
&\leq~ \biggl(\sum_{i=1}^n a_i^{1+t} b_i^{1-t}\biggr) \biggl(\sum_{i=1}^n a_i^{1-t} b_i^{1+t}\biggr)
~\leq~ \biggl(\sum_{i=1}^n a_i^2\biggr) \biggl(\sum_{i=1}^n b_i^2\biggr).
\end{align}</math>
}}

This theorem can be deduced from [[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 journal|last1=Moslehian|first1=M.S.|last2=Matharu|first2=J.S.|last3=Aujla|first3=J.S.|date=2011|title=Non-commutative Callebaut inequality|journal=Linear Algebra and Its Applications|volume=436|issue=9|pages=3347–3353|doi=10.1016/j.laa.2011.11.024|arxiv=1112.3003|s2cid=119592971}}</ref>

Several matrix versions of the Cauchy–Schwarz inequality and [[Kantorovich inequality]] are applied to linear regression models.<ref>
{{cite journal |author1=Liu, Shuangzhe|author2=Neudecker, Heinz 
|year=1999
|title= A survey of Cauchy–Schwarz and Kantorovich-type matrix inequalities
|journal= Statistical Papers |volume=40 |pages=55–73 
|doi=10.1007/BF02927110 
|s2cid=122719088 
}}
</ref>
<ref>{{Cite journal| last1=Liu|first1=Shuangzhe|
last2= Trenkler|first2=Götz| last3=Kollo|first3=Tõnu| 
last4=von Rosen|first4=Dietrich| 
last5=Baksalary|first5=Oskar Maria| 
date= 2023|
title= Professor Heinz Neudecker and matrix differential calculus| 
journal= Statistical Papers|volume=65 |issue=4 |pages=2605–2639 | language=en|
doi= 10.1007/s00362-023-01499-w|s2cid=263661094 }}</ref>

== See also ==

* {{annotated link|Bessel's inequality}}
* {{annotated link|Hölder's inequality}}
* {{annotated link|Jensen's inequality}}
* {{annotated link|Kantorovich inequality}}
* {{annotated link|Kunita–Watanabe inequality}}
* {{annotated link|Minkowski inequality}}
* {{annotated link|Paley–Zygmund inequality}}

== Notes ==

{{reflist|group=note}}
{{reflist|group=proof}}

== Citations ==
{{reflist|30em}}

== References ==

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* {{Halmos A Hilbert Space Problem Book 1982}} <!-- {{sfn|Halmos|1982|pp=}} -->
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{{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.

{{Lp spaces}}
{{Functional Analysis}}
{{HilbertSpace}}

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