Editing Cauchy–Schwarz inequality (section)

== 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>