Editing Cauchy–Schwarz inequality (section)

== 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]].