Cauchy–Schwarz inequality

The Cauchy–Schwarz inequality states that for all vectors $\backslash mathbf\{u\}$ and $\backslash mathbf\{v\}$ of an inner product space

{{NumBlk|:|$\backslash left\; |\backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{v\}\; \backslash rangle\backslash right\; |^2\; \backslash leq\; \backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{u\}\; \backslash rangle\; \backslash cdot\; \backslash langle\; \backslash mathbf\{v\},\; \backslash mathbf\{v\}\; \backslash rangle,$|{{EquationRef|Cauchy–Schwarz inequality [written using only the inner product]|1}}}}

where $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ is the inner product. Examples of inner products include the real and complex dot product; see the examples in inner product. Every inner product gives rise to a Euclidean $\backslash ell\_2$ norm, called the {{em|canonical}} or inner product space#Norm, where the norm of a vector $\backslash mathbf\{u\}$ is denoted and defined by

$$\backslash |\backslash mathbf\{u\}\backslash |\; :=\; \backslash sqrt\{\backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{u\}\; \backslash rangle\},$$ where $\backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{u\}\; \backslash rangle$ 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:{{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}}{{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}}

{{NumBlk|:|$|\backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{v\}\; \backslash rangle|\; \backslash leq\; \backslash |\backslash mathbf\{u\}\backslash |\; \backslash |\backslash mathbf\{v\}\backslash |.$|{{EquationRef|Cauchy–Schwarz inequality - written using norm and inner product|2}}}}

Moreover, the two sides are equal if and only if $\backslash mathbf\{u\}$ and $\backslash mathbf\{v\}$ are linearly dependent.{{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}}{{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  = 0 or {{pipe}}c> = 0. From the definition of {{pipe}}c>, we conclude that {{pipe}}a> and {{pipe}}b> must be proportional.}}{{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.}}

Sedrakyan's inequality, also known as Bergström's inequality, Engel's form, Titu's lemma (or the T2 lemma), states that for real numbers $u\_1,\; u\_2,\; \backslash dots,\; u\_n$ and positive real numbers $v\_1,\; v\_2,\; \backslash dots,\; v\_n$:

$$\backslash frac\{\backslash left(u\_1\; +\; u\_2\; +\; \backslash cdots\; +\; u\_n\backslash right)^2\}\{v\_1\; +\; v\_2\; +\; \backslash cdots\; +\; v\_n\}\; \backslash leq\; \backslash frac\{u^2\_1\}\{v\_1\}\; +\; \backslash frac\{u^2\_2\}\{v\_2\}\; +\; \backslash cdots\; +\; \backslash frac\{u^2\_n\}\{v\_n\},$$

or, using summation notation,

$$\backslash biggl(\backslash sum\_\{i=1\}^n\; u\_i\backslash biggr)^2\; \backslash bigg/\; \backslash sum\_\{i=1\}^n\; v\_i\; \backslash ,\backslash leq\backslash ,\; \backslash sum\_\{i=1\}^n\; \backslash frac\{u\_i^2\}\{v\_i\}.$$

It is a direct consequence of the Cauchy–Schwarz inequality, obtained by using the dot product on $\backslash R^n$ upon substituting $u\_i\text{'}\; =\; \backslash frac\{u\_i\}\{\backslash sqrt\{v\_i\backslash vphantom\{t\}\}\}$ and $v\_i\text{'}\; =\; \{\backslash textstyle\; \backslash sqrt\{v\_i\backslash vphantom\{t\}\}\}$. This form is especially helpful when the inequality involves fractions where the numerator is a perfect square.

File:Cauchy-Schwarz inequation in Euclidean plane.gif

The real vector space $\backslash R^2$ denotes the 2-dimensional plane. It is also the 2-dimensional Euclidean space where the inner product is the dot product.

If $\backslash mathbf\{u\}\; =\; (u\_1,\; u\_2)$ and $\backslash mathbf\{v\}\; =\; (v\_1,\; v\_2)$ then the Cauchy–Schwarz inequality becomes:

$$\backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{v\}\; \backslash rangle^2\; =\; \backslash bigl(\backslash |\backslash mathbf\{u\}\backslash |\; \backslash |\backslash mathbf\{v\}\backslash |\; \backslash cos\; \backslash theta\backslash bigr)^2\; \backslash leq\; \backslash |\backslash mathbf\{u\}\backslash |^2\; \backslash |\backslash mathbf\{v\}\backslash |^2,$$

where $\backslash theta$ is the angle between $\backslash mathbf\{u\}$ and $\backslash mathbf\{v\}$.

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 $u\_1$, $u\_2$, $v\_1$, and $v\_2$ as

$$\backslash left(u\_1\; v\_1\; +\; u\_2\; v\_2\backslash right)^2\; \backslash leq\; \backslash left(u\_1^2\; +\; u\_2^2\backslash right)\; \backslash left(v\_1^2\; +\; v\_2^2\backslash right),$$

where equality holds if and only if the vector $\backslash left(u\_1,\; u\_2\backslash right)$ is in the same or opposite direction as the vector $\backslash left(v\_1,\; v\_2\backslash right)$, or if one of them is the zero vector.

{{anchor|real number proof}}In Euclidean space $\backslash R^n$ with the standard inner product, which is the dot product, the Cauchy–Schwarz inequality becomes:

$$\backslash biggl(\backslash sum\_\{i=1\}^n\; u\_i\; v\_i\backslash biggr)^2\; \backslash leq\; \backslash biggl(\backslash sum\_\{i=1\}^n\; u\_i^2\backslash biggr)\; \backslash biggl(\backslash sum\_\{i=1\}^n\; v\_i^2\backslash biggr).$$

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

$$\backslash tfrac\{1\}\{2\}\; \backslash sum\_\{i=1\}^n\backslash sum\_\{j=1\}^n\; (u\_i\; v\_j\; -\; u\_j\; v\_i)^2\; \backslash ge\; 0$$

or by considering the following quadratic polynomial in $x$

$$(u\_1\; x\; +\; v\_1)^2\; +\; \backslash cdots\; +\; (u\_n\; x\; +\; v\_n)^2\; =\; \backslash biggl(\backslash sum\_i\; u\_i^2\backslash biggr)\; x^2\; +\; 2\; \backslash biggl(\backslash sum\_i\; u\_i\; v\_i\backslash biggr)\; x\; +\; \backslash sum\_i\; v\_i^2.$$

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,

$$\backslash biggl(\backslash sum\_i\; u\_i\; v\_i\backslash biggr)^2\; -\; \backslash biggl(\backslash sum\_i\; \{u\_i^2\}\backslash biggr)\; \backslash biggl(\backslash sum\_i\; \{v\_i^2\}\backslash biggr)\; \backslash leq\; 0.$$

If $\backslash mathbf\{u\},\; \backslash mathbf\{v\}\; \backslash in\; \backslash Complex^n$ with $\backslash mathbf\{u\}\; =\; (u\_1,\; \backslash ldots,\; u\_n)$ and $\backslash mathbf\{v\}\; =\; (v\_1,\; \backslash ldots,\; v\_n)$ (where $u\_1,\; \backslash ldots,\; u\_n\; \backslash in\; \backslash Complex$ and $v\_1,\; \backslash ldots,\; v\_n\; \backslash in\; \backslash Complex$) and if the inner product on the vector space $\backslash Complex^n$ is the canonical complex inner product (defined by $\backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{v\}\; \backslash rangle\; :=\; u\_1\; \backslash overline\{v\_1\}\; +\; \backslash cdots\; +\; u\_\{n\}\; \backslash overline\{v\_n\},$ where the bar notation is used for complex conjugation), then the inequality may be restated more explicitly as follows:

$$\backslash bigl|\backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{v\}\; \backslash rangle\backslash bigr|^2$$

= \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.

That is,

$$\backslash bigl|u\_1\; \backslash bar\{v\}\_1\; +\; \backslash cdots\; +\; u\_n\; \backslash bar\{v\}\_n\backslash bigr|^2\; \backslash leq\; \backslash bigl(|u\_1|\{\}^2\; +\; \backslash cdots\; +\; |u\_n|\{\}^2\backslash bigr)\; \backslash bigl(|v\_1|\{\}^2\; +\; \backslash cdots\; +\; |v\_n|\{\}^2\backslash bigr).$$

For the inner product space of square-integrable complex-valued functions, the following inequality holds.

$$\backslash left|\backslash int\_\{\backslash R^n\}\; f(x)\; \backslash overline\{g(x)\}\backslash ,dx\backslash right|^2\; \backslash leq\; \backslash int\_\{\backslash R^n\}\; \backslash bigl|f(x)\backslash bigr|^2\backslash ,dx\; \backslash int\_\{\backslash R^n\}\; \backslash bigl|g(x)\backslash bigr|^2\; \backslash ,dx.$$

The Hölder inequality is a generalization of this.

In any inner product space, the triangle inequality is a consequence of the Cauchy–Schwarz inequality, as is now shown:

$$\backslash begin\{alignat\}\{4\}$$

\|\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}

Taking square roots gives the triangle inequality:

$$\backslash |\backslash mathbf\{u\}\; +\; \backslash mathbf\{v\}\backslash |\; \backslash leq\; \backslash |\backslash mathbf\{u\}\backslash |\; +\; \backslash |\backslash mathbf\{v\}\backslash |.$$

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.{{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}}{{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}}

The Cauchy–Schwarz inequality allows one to extend the notion of "angle between two vectors" to any real inner-product space by defining:{{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}}{{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}}

$$\backslash cos\backslash theta\_\{\backslash mathbf\{u\}\; \backslash mathbf\{v\}\}\; =\; \backslash frac\{\backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{v\}\; \backslash rangle\}\{\backslash |\backslash mathbf\{u\}\backslash |\; \backslash |\backslash mathbf\{v\}\backslash |\}.$$

The Cauchy–Schwarz inequality proves that this definition is sensible, by showing that the right-hand side lies in the interval {{math|[−1, 1]}} and justifies the notion that (real) Hilbert spaces are simply generalizations of the Euclidean space. It can also be used to define an angle in complex inner-product spaces, by taking the absolute value or the real part of the right-hand side,{{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}}{{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}} as is done when extracting a metric from quantum fidelity.

Let $X$ and $Y$ be random variables. Then the covariance inequality{{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}}{{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}} is given by:

$$\backslash operatorname\{Var\}(X)\; \backslash geq\; \backslash frac\{\backslash operatorname\{Cov\}(X,\; Y)^2\}\{\backslash operatorname\{Var\}(Y)\}.$$

After defining an inner product on the set of random variables using the expectation of their product,

$$\backslash langle\; X,\; Y\; \backslash rangle\; :=\; \backslash operatorname\{E\}(X\; Y),$$

the Cauchy–Schwarz inequality becomes

$$\backslash bigl|\backslash operatorname\{E\}(XY)\backslash bigr|^2\; \backslash leq\; \backslash operatorname\{E\}(X^2)\; \backslash operatorname\{E\}(Y^2).$$

To prove the covariance inequality using the Cauchy–Schwarz inequality, let $\backslash mu\; =\; \backslash operatorname\{E\}(X)$ and $\backslash nu\; =\; \backslash operatorname\{E\}(Y),$ then

$$\backslash begin\{align\}$$

\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}

where $\backslash operatorname\{Var\}$ denotes variance and $\backslash operatorname\{Cov\}$ denotes covariance.

There are many different proofs{{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}} of the Cauchy–Schwarz inequality other than those given below.

When consulting other sources, there are often two sources of confusion. First, some authors define {{math|⟨⋅,⋅⟩}} to be linear in the second argument rather than the first.

Second, some proofs are only valid when the field is $\backslash mathbb\; R$ and not $\backslash mathbb\; C.${{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}}

This section gives two proofs of the following theorem:

{{math theorem|name=Cauchy–Schwarz inequality|note=|style=|math_statement=

Let $\backslash mathbf\{u\}$ and $\backslash mathbf\{v\}$ be arbitrary vectors in an inner product space over the scalar field $\backslash mathbb\{F\},$ where $\backslash mathbb\; F$ is the field of real numbers $\backslash R$ or complex numbers $\backslash Complex.$ Then

{{NumBlk|:|$\backslash bigl|\backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{v\}\; \backslash rangle\backslash bigr|\; \backslash leq\; \backslash |\backslash mathbf\{u\}\backslash |\; \backslash |\backslash mathbf\{v\}\backslash |$|{{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 $\backslash mathbf\{u\}$ and $\backslash mathbf\{v\}$ are linearly dependent.

Moreover, if $\backslash left|\; \backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{v\}\; \backslash rangle\backslash right|\; =\; \backslash |\backslash mathbf\{u\}\backslash |\; \backslash |\backslash mathbf\{v\}\backslash |$ and $\backslash mathbf\{v\}\; \backslash neq\; \backslash mathbf\{0\}$ then $\backslash mathbf\{u\}\; =\; \backslash frac\{\backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{v\}\; \backslash rangle\}\{\backslash |\backslash mathbf\{v\}\backslash |^2\}\; \backslash mathbf\{v\}.$

}}

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 $\backslash |\backslash mathbf\{u\}\backslash |\backslash |\backslash mathbf\{v\}\backslash |=\; 0$) is the same. It is presented immediately below only once to reduce repetition. It also includes the easy part of the proof of the Equality Characterization given above; that is, it proves that if $\backslash mathbf\{u\}$ and $\backslash mathbf\{v\}$ are linearly dependent then $\backslash bigl|\backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{v\}\; \backslash rangle\backslash bigr|\; =\; \backslash |\backslash mathbf\{u\}\backslash |\; \backslash |\backslash mathbf\{v\}\backslash |.$

{{collapse top|title=Proof of the trivial parts: Case where a vector is $\backslash mathbf\{0\}$ and also one direction of the Equality Characterization|left=true}}

By definition, $\backslash mathbf\{u\}$ and $\backslash mathbf\{v\}$ are linearly dependent if and only if one is a scalar multiple of the other.

If $\backslash mathbf\{u\}\; =\; c\; \backslash mathbf\{v\}$ where $c$ is some scalar then

$$|\backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{v\}\; \backslash rangle|$$

= |\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}\|

which shows that equality holds in the {{EquationNote|Cauchy–Schwarz Inequality}}.

The case where $\backslash mathbf\{v\}\; =\; c\; \backslash mathbf\{u\}$ for some scalar $c$ follows from the previous case:

$$|\backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{v\}\; \backslash rangle|$$

= |\langle \mathbf{v}, \mathbf{u} \rangle|

=\|\mathbf{v}\| \|\mathbf{u}\|.

In particular, if at least one of $\backslash mathbf\{u\}$ and $\backslash mathbf\{v\}$ is the zero vector then $\backslash mathbf\{u\}$ and $\backslash mathbf\{v\}$ are necessarily linearly dependent (for example, if $\backslash mathbf\{u\}\; =\; \backslash mathbf\{0\}$ then $\backslash mathbf\{u\}\; =\; c\; \backslash mathbf\{v\}$ where $c\; =\; 0$), 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 Equality Characterization must be shown.

The special case of $\backslash mathbf\{v\}\; =\; \backslash mathbf\{0\}$ was proven above so it is henceforth assumed that $\backslash mathbf\{v\}\; \backslash neq\; \backslash mathbf\{0\}.$

Let

$$\backslash mathbf\{z\}\; :=\; \backslash mathbf\{u\}\; -\; \backslash frac\; \{\backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{v\}\; \backslash rangle\}\; \{\backslash langle\; \backslash mathbf\{v\},\; \backslash mathbf\{v\}\; \backslash rangle\}\; \backslash mathbf\{v\}.$$

It follows from the linearity of the inner product in its first argument that:

$$\backslash langle\; \backslash mathbf\{z\},\; \backslash mathbf\{v\}\; \backslash rangle$$

= \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.

Therefore, $\backslash mathbf\{z\}$ is a vector orthogonal to the vector $\backslash mathbf\{v\}$ (Indeed, $\backslash mathbf\{z\}$ is the projection of $\backslash mathbf\{u\}$ onto the plane orthogonal to $\backslash mathbf\{v\}.$) We can thus apply the Pythagorean theorem to

$$\backslash mathbf\{u\}=\; \backslash frac\{\backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{v\}\; \backslash rangle\}\; \{\backslash langle\; \backslash mathbf\{v\},\; \backslash mathbf\{v\}\; \backslash rangle\}\; \backslash mathbf\{v\}\; +\; \backslash mathbf\{z\}$$

which gives

$$\backslash |\backslash mathbf\{u\}\backslash |^2$$

= \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}.

The Cauchy–Schwarz inequality follows by multiplying by $\backslash |\backslash mathbf\{v\}\backslash |^2$ and then taking the square root.

Moreover, if the relation $\backslash geq$ in the above expression is actually an equality, then $\backslash |\backslash mathbf\{z\}\backslash |^2\; =\; 0$ and hence $\backslash mathbf\{z\}\; =\; \backslash mathbf\{0\};$ the definition of $\backslash mathbf\{z\}$ then establishes a relation of linear dependence between $\backslash mathbf\{u\}$ and $\backslash mathbf\{v\}.$ The converse was proved at the beginning of this section, so the proof is complete. $\backslash blacksquare$

Consider an arbitrary pair of vectors $\backslash mathbf\{u\},\; \backslash mathbf\{v\}$. Define the function $p\; :\; \backslash R\; \backslash to\; \backslash R$ defined by $p(t)\; =\; \backslash langle\; t\backslash alpha\backslash mathbf\{u\}\; +\; \backslash mathbf\{v\},\; t\backslash alpha\backslash mathbf\{u\}\; +\; \backslash mathbf\{v\}\backslash rangle$, where $\backslash alpha$ is a complex number satisfying $|\backslash alpha|\; =\; 1$ and $\backslash alpha\backslash langle\backslash mathbf\{u\},\; \backslash mathbf\{v\}\backslash rangle\; =\; |\backslash langle\backslash mathbf\{u\},\; \backslash mathbf\{v\}\backslash rangle|$.

Such an $\backslash alpha$ exists since if $\backslash langle\backslash mathbf\{u\},\; \backslash mathbf\{v\}\backslash rangle\; =\; 0$ then $\backslash alpha$ can be taken to be 1.

Since the inner product is positive-definite, $p(t)$ only takes non-negative real values. On the other hand, $p(t)$ can be expanded using the bilinearity of the inner product:

$$\backslash begin\{align\}$$

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}

Thus, $p$ is a polynomial of degree $2$ (unless $\backslash mathbf\{u\}\; =\; 0,$ which is a case that was checked earlier). Since the sign of $p$ does not change, the discriminant of this polynomial must be non-positive:

$$\backslash Delta\; =\; 4\; \backslash bigl(\backslash ,|\backslash langle\; \backslash mathbf\{u\},\; \backslash mathbf\{v\}\; \backslash rangle|^2\; -\; \backslash Vert\; \backslash mathbf\{u\}\; \backslash Vert^2\; \backslash Vert\; \backslash mathbf\{v\}\; \backslash Vert^2\backslash bigr)\; \backslash leq\; 0.$$

The conclusion follows.{{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}}

For the equality case, notice that $\backslash Delta\; =\; 0$ happens if and only if $p(t)\; =\; \backslash bigl(t\backslash Vert\; \backslash mathbf\{u\}\; \backslash Vert\; +\; \backslash Vert\; \backslash mathbf\{v\}\; \backslash Vert\backslash bigr)^2.$ If $t\_0\; =\; -\backslash Vert\; \backslash mathbf\{v\}\; \backslash Vert\; /\; \backslash Vert\; \backslash mathbf\{u\}\; \backslash Vert,$ then $p(t\_0)\; =\; \backslash langle\; t\_0\backslash alpha\backslash mathbf\{u\}\; +\; \backslash mathbf\{v\},t\_0\backslash alpha\backslash mathbf\{u\}\; +\; \backslash mathbf\{v\}\backslash rangle\; =\; 0,$ and hence $\backslash mathbf\{v\}\; =\; -t\_0\backslash alpha\backslash mathbf\{u\}.$

Various generalizations of the Cauchy–Schwarz inequality exist. Hölder's inequality generalizes it to $L^p$ 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 algebras, 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 $L^2(m),\; m$ being a finite measure, the standard inner product gives rise to a positive functional $\backslash varphi$ by $\backslash varphi\; (g)\; =\; \backslash langle\; g,\; 1\; \backslash rangle.$ Conversely, every positive linear functional $\backslash varphi$ on $L^2(m)$ can be used to define an inner product $\backslash langle\; f,\; g\; \backslash rangle\; \_\backslash varphi\; :=\; \backslash varphi\backslash left(g^*\; f\backslash right),$ where $g^*$ is the pointwise complex conjugate of $g.$ In this language, the Cauchy–Schwarz inequality becomes{{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}}

$$\backslash bigl|\backslash varphi(g^*\; f)\backslash bigr|^2\; \backslash leq\; \backslash varphi\backslash left(f^*\; f\backslash right)\; \backslash varphi\backslash left(g^*\; g\backslash right),$$

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

{{math theorem|name=Cauchy–Schwarz inequality for positive functionals on C*-algebras{{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}}{{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}}|note=|style=|math_statement=

If $\backslash varphi$ is a positive linear functional on a C*-algebra $A,$ then for all $a,\; b\; \backslash in\; A,$ $\backslash left|\backslash varphi\backslash left(b^*a\backslash right)\backslash right|^2\; \backslash leq\; \backslash varphi\backslash left(b^*b\backslash right)\; \backslash varphi\backslash left(a^*a\backslash right).$

}}

The next two theorems are further examples in operator algebra.

{{math theorem|name=Kadison–Schwarz inequality{{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}}{{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}}|note=Named after Richard Kadison|style=|math_statement=

If $\backslash varphi$ is a unital positive map, then for every normal element $a$ in its domain, we have $\backslash varphi(a^*a)\; \backslash geq\; \backslash varphi\backslash left(a^*\backslash right)\; \backslash varphi(a)$ and $\backslash varphi\backslash left(a^*a\backslash right)\; \backslash geq\; \backslash varphi(a)\; \backslash varphi\backslash left(a^*\backslash right).$

}}

This extends the fact $\backslash varphi\backslash left(a^*a\backslash right)\; \backslash cdot\; 1\; \backslash geq\; \backslash varphi(a)^*\; \backslash varphi(a)\; =\; |\backslash varphi(a)|^2,$ when $\backslash varphi$ is a linear functional. The case when $a$ is self-adjoint, that is, $a\; =\; a^*,$ is sometimes known as Kadison's inequality.

{{math theorem|name=Cauchy–Schwarz inequality|note=Modified Schwarz inequality for 2-positive maps{{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}}|style=|math_statement=

For a 2-positive map $\backslash varphi$ between C*-algebras, for all $a,\; b$ in its domain,

$$\backslash begin\{align\}$$

\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}

}}

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

{{math theorem|name=Callebaut's Inequality{{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}}|note=|style=|math_statement=

For reals $0\; \backslash leq\; s\; \backslash leq\; t\; \backslash leq\; 1,$

$$\backslash begin\{align\}$$

\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}

}}

This theorem can be deduced from Hölder's inequality.{{cite book|title=Callebaut's inequality|publisher=Entry in the AoPS Wiki|url=https://artofproblemsolving.com/wiki/index.php?title=Callebaut%27s_Inequality}} There are also non-commutative versions for operators and tensor products of matrices.{{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}}

Several matrix versions of the Cauchy–Schwarz inequality and Kantorovich inequality are applied to linear regression models.

{{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

}}

{{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 }}

• {{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}}

{{reflist|group=note}}

{{reflist|group=proof}}

{{reflist|30em}}

{{refbegin}}

• {{citation|first1=J. M.|last1=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|s2cid=122631202 |url=http://projecteuclid.org/euclid.afa/1429286046}}
• {{citation|first=Viktor|last=Bunyakovsky|author-link=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=6|url=http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/bunyakovsky.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/bunyakovsky.pdf |archive-date=2022-10-09 |url-status=live}}
• {{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|archive-url=https://web.archive.org/web/20080720034744/http://jipam.vu.edu.au/article.php?sid=301|archive-date=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|doi-access=free }}
• {{Halmos A Hilbert Space Problem Book 1982}}
• {{citation|first=R. V.|last=Kadison|author-link=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 format|url=http://www.mediafire.com/file/1mw1tkgozzu|year=1982}}
• {{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 |archive-url=https://ghostarchive.org/archive/20221009/http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/Schwarz.pdf |archive-date=2022-10-09 |url-status=live}}
• {{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}}

• [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.

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