Cauchy–Schwarz inequality explained

The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality)^{[1] [2] [3]} is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics.^[4]

Inner products of vectors can describe finite sums (via finite-dimensional vector spaces), infinite series (via vectors in sequence spaces), and integrals (via vectors in Hilbert spaces). The inequality for sums was published by . The corresponding inequality for integrals was published by and . Schwarz gave the modern proof of the integral version.

Statement of the inequality

The Cauchy–Schwarz inequality states that for all vectors

u and v of an inner product space

where

\langle ⋅ , ⋅ \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 l _{2} norm, called the or, where the norm of a vector u is denoted and defined by$$\backslash |\backslash mathbf\backslash |\; :=\; \backslash sqrt,$$ where \langle u , u \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:^{[5] [6]}

Moreover, the two sides are equal if and only if

u and v are linearly dependent.^{[7] [8] [9]}

Special cases

Sedrakyan's lemma - Positive real numbers

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,} ... , u _{n} and positive real numbers v _{1,} v _{2,} ... , v _{n} :$$\backslash frac\; \backslash leq\; \backslash sum\_^n\; \backslash frac\; \backslash quad\; \backslash text\; \backslash quad\; \backslash frac\; \backslash leq\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots\; +\; \backslash frac\; .$$

It is a direct consequence of the Cauchy–Schwarz inequality, obtained by using the dot product on

\R ^{n} upon substituting u _{i'} = (u _{i})/(\sqrt{v _{i}) } and v _{i'} = \sqrt{v _{i}} . This form is especially helpful when the inequality involves fractions where the numerator is a perfect square.

- The plane

The real vector space

\R ^{2} denotes the 2-dimensional plane. It is also the 2-dimensional Euclidean space where the inner product is the dot product. If u = \left(u _{1,} u _{2\right)} and v = \left(v _{1,} v _{2\right)} then the Cauchy–Schwarz inequality becomes:$$\backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle^2\; =\; (\backslash |\backslash mathbf\backslash |\; \backslash |\backslash mathbf\backslash |\; \backslash cos\; \backslash theta)^2\; \backslash leq\; \backslash |\backslash mathbf\backslash |^2\; \backslash |\backslash mathbf\backslash |^2,$$where \theta is the angle between u and 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 \left(u _{1,} u _{2\right)} is in the same or opposite direction as the vector \left(v _{1,} v _{2\right)} , or if one of them is the zero vector.

: n-dimensional Euclidean space

\R ^{n} with the standard inner product, which is the dot product, the Cauchy–Schwarz inequality becomes:$$\backslash left(\backslash sum\_^n\; u\_i\; v\_i\backslash right)^2\; \backslash leq\; \backslash left(\backslash sum\_^n\; u\_i^2\backslash right)\; \backslash left(\backslash sum\_^n\; v\_i^2\backslash right).$$

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 $$\backslash frac\; \backslash sum\_\; (u\_i\; v\_j\; -\; u\_j\; v\_i)^2\; \backslash ge\; 0$$

or by considering the following quadratic polynomial in

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

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,$$\backslash left(\backslash sum\_i\; u\_i\; v\_i\backslash right)^2\; -\; \backslash left(\backslash sum\_i\; \backslash right)\; \backslash left(\backslash sum\_i\; \backslash right)\; \backslash leq\; 0.$$

: n-dimensional Complex space

If

u , v \in \Complex ^{n} with u = \left(u _{1,} \ldots, u _{n\right)} and v = \left(v _{1,} \ldots, v _{n\right)} (where u _{1,} \ldots, u _{n} \in \Complex and v _{1,} \ldots, v _{n} \in \Complex ) and if the inner product on the vector space \Complex ^{n} is the canonical complex inner product (defined by \langle u , v \rangle : = u _{1} \overline{v _{1}} + … + u _{n} \overline{v _{n},} where the bar notation is used for complex conjugation), then the inequality may be restated more explicitly as follows: $$|\backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle|^2\; =\; \backslash left|\backslash sum\_^n\; u\_k\backslash bar\_k\backslash right|^2\; \backslash leq\; \backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle\; \backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle\; =\; \backslash left(\backslash sum\_^n\; u\_k\; \backslash bar\_k\backslash right)\; \backslash left(\backslash sum\_^n\; v\_k\; \backslash bar\_k\backslash right)\; =\; \backslash sum\_^n\; \backslash left|u\_j\backslash right|^2\; \backslash sum\_^n\; \backslash left|v\_k\backslash right|^2.$$

That is,$$\backslash left|u\_1\; \backslash bar\_1\; +\; \backslash vdots\; +\; u\_n\; \backslash bar\_n\backslash right|^2\; \backslash leq\; \backslash left(\backslash left|u\_1\backslash right|^2\; +\; \backslash cdots\; +\; \backslash left|u\_n\backslash right|^2\backslash right)\; \backslash left(\backslash left|v\_1\backslash right|^2\; +\; \backslash cdots\; +\; \backslash left|v\_n\backslash right|^2\backslash right).$$

For the inner product space of square-integrable complex-valued functions, the following inequality holds.$$\backslash left|\backslash int\_\; f(x)\; \backslash overline\backslash ,dx\backslash right|^2\; \backslash leq\; \backslash int\_\; |f(x)|^2\backslash ,dx\; \backslash int\_\; |g(x)|^2\; \backslash ,dx.$$

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:

Taking square roots gives the triangle inequality:$$\backslash |\backslash mathbf\; +\; \backslash mathbf\backslash |\; \backslash leq\; \backslash |\backslash mathbf\backslash |\; +\; \backslash |\backslash mathbf\backslash |.$$

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.^{[10] [11]}

Geometry

The Cauchy–Schwarz inequality allows one to extend the notion of "angle between two vectors" to any real inner-product space by defining:^{[12] [13]} $$\backslash cos\backslash theta\_\; =\; \backslash frac.$$

The Cauchy–Schwarz inequality proves that this definition is sensible, by showing that the right-hand side lies in the interval 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,^{[14] [15]} as is done when extracting a metric from quantum fidelity.

Probability theory

Let

X and Y be random variables. Then the covariance inequality^{[16] [17]} is given by:$$\backslash operatorname(X)\; \backslash geq\; \backslash frac.$$

After defining an inner product on the set of random variables using the expectation of their product,$$\backslash langle\; X,\; Y\; \backslash rangle\; :=\; \backslash operatorname(X\; Y),$$the Cauchy–Schwarz inequality becomes$$|\backslash operatorname(XY)|^2\; \backslash leq\; \backslash operatorname(X^2)\; \backslash operatorname(Y^2).$$

To prove the covariance inequality using the Cauchy–Schwarz inequality, let

\mu = \operatorname{E}(X) and \nu = \operatorname{E}(Y), thenwhere \operatorname{Var} denotes variance and \operatorname{Cov} denotes covariance.

Proofs

There are many different proofs^[18] 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 to be linear in the second argument rather than the first. Second, some proofs are only valid when the field is

R and not C. ^[19]

This section gives two proofs of the following theorem:

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

\| u \|\| v \| = 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 u and v are linearly dependent then \left|\langle u , v \rangle\right| = \| u \| \| v \|.

By definition,

u and v are linearly dependent if and only if one is a scalar multiple of the other. If u = c v where c is some scalar then $$|\backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle|\; =\; |\backslash langle\; c\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle|\; =\; |c\; \backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle|\; =\; |c|\backslash |\backslash mathbf\backslash |\; \backslash |\backslash mathbf\backslash |=\backslash |c\; \backslash mathbf\backslash |\; \backslash |\backslash mathbf\backslash |=\backslash |\backslash mathbf\backslash |\; \backslash |\backslash mathbf\backslash |$$

which shows that equality holds in the . The case where

v = c u for some scalar c follows from the previous case:$$|\backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle|\; =\; |\backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle|=\backslash |\backslash mathbf\backslash |\; \backslash |\backslash mathbf\backslash |.$$

In particular, if at least one of

u and v is the zero vector then u and v are necessarily linearly dependent (for example, if u = 0 then u = c v where c = 0 ), so the above computation shows that the Cauchy-Schwarz inequality holds in this case.

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.

Proof via the Pythagorean theorem

The special case of

v = 0 was proven above so it is henceforth assumed that v ≠ 0 . Let$$\backslash mathbf\; :=\; \backslash mathbf\; -\; \backslash frac\; \backslash mathbf.$$

It follows from the linearity of the inner product in its first argument that:$$\backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle\; =\; \backslash left\backslash langle\; \backslash mathbf\; -\; \backslash frac\; \backslash mathbf,\; \backslash mathbf\; \backslash right\backslash rangle\; =\; \backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle\; -\; \backslash frac\; \backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle\; =\; 0.$$

Therefore,

z is a vector orthogonal to the vector v (Indeed, z is the projection of u onto the plane orthogonal to v . ) We can thus apply the Pythagorean theorem to$$\backslash mathbf=\; \backslash frac\; \backslash mathbf\; +\; \backslash mathbf$$which gives$$\backslash |\backslash mathbf\backslash |^2\; =\; \backslash left|\backslash frac\backslash right|^2\; \backslash |\backslash mathbf\backslash |^2\; +\; \backslash |\backslash mathbf\backslash |^2\; =\; \backslash frac$$

^2

\,\|\mathbf\|^2 + \|\mathbf\|^2 = \frac

^2

+ \|\mathbf\|^2 \geq \frac

^2

.

The Cauchy–Schwarz inequality follows by multiplying by

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

Proof by analyzing a quadratic

Consider an arbitrary pair of vectors

u , v . Define the function p : \R \to \R defined by p(t) = \langle t\alpha u + v , t\alpha u + v \rangle , where \alpha is a complex number satisfying |\alpha| = 1 and \alpha\langle u , v \rangle = |\langle u , v \rangle| .Such an \alpha exists since if \langle u , v \rangle = 0 then \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:Thus, p is a polynomial of degree 2 (unless 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:$$\backslash Delta\; =\; 4\; \backslash left(|\backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle|^2\; -\; \backslash Vert\; \backslash mathbf\; \backslash Vert^2\; \backslash Vert\; \backslash mathbf\; \backslash Vert^2\backslash right)\; \backslash leqslant\; 0.$$The conclusion follows.^[20]

For the equality case, notice that

\Delta = 0 happens if and only if p(t) = (t\Vert u \Vert + \Vert v \Vert) ^{2.} If t _{0} = - \Vert v \Vert / \Vert u \Vert, then p(t _{0)} = \langle t _{0\alphau} + v ,t _{0\alphau} + v \rangle = 0 , and hence v = - t _{0\alphau.}

Generalizations

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 \varphi by \varphi (g) = \langle g, 1 \rangle. Conversely, every positive linear functional \varphi on L ^{2(m)} can be used to define an inner product \langle f, g \rangle _{\varphi} : = \varphi\left(g ^{*} f\right), where g ^{*} is the pointwise complex conjugate of g. In this language, the Cauchy–Schwarz inequality becomes^[21] $$\backslash left|\backslash varphi\backslash left(g^*\; f\backslash right)\backslash right|^2\; \backslash leq\; \backslash varphi\backslash left(f^*\; f\backslash right)\; \backslash varphi\backslash left(g^*\; g\backslash right),$$

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

The next two theorems are further examples in operator algebra.

This extends the fact

\varphi\left(a ^{*a\right)} ⋅ 1 \geq \varphi(a) ^{*} \varphi(a) = |\varphi(a)| ^{2,} when \varphi is a linear functional. The case when a is self-adjoint, that is, a = a ^{*,} is sometimes known as Kadison's inequality.

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

This theorem can be deduced from Hölder's inequality.^[22] There are also non-commutative versions for operators and tensor products of matrices.^[23]

Several matrix versions of the Cauchy–Schwarz inequality and Kantorovich inequality are applied to linear regression models.^{[24] [25]}

References

• • • • • • • .
  • .

External links

• Earliest Uses: The entry on the Cauchy–Schwarz inequality has some historical information.
• Example of application of Cauchy–Schwarz inequality to determine Linearly Independent Vectors Tutorial and Interactive program.

Notes and References

1. Web site: Hermann Amandus Schwarz. J.J.. O'Connor. E.F.. Robertson. University of St Andrews, Scotland .
2. Web site: Cauchy-Bunyakovsky-Schwarz inequality. Branko. Ćurgus. Western Washington University. Department of Mathematics .
3. Web site: Cauchy's inequality. https://ghostarchive.org/archive/20221009/https://mathcs.clarku.edu/~djoyce/ma130/cauchy.pdf . 2022-10-09 . live. David E.. Joyce. Clark University. Department of Mathematics and Computer Science .
4. Book: Steele, J. Michael. 2004. The Cauchy–Schwarz Master Class: an Introduction to the Art of Mathematical Inequalities. The Mathematical Association of America. 978-0521546775. 1. ...there is no doubt that this is one of the most widely used and most important inequalities in all of mathematics..
5. Book: Strang, Gilbert. 19 July 2005. Linear Algebra and its Applications. 4th. 3.2. Cengage Learning. Stamford, CT. 978-0030105678. 154–155.
6. Book: Hunter. John K.. Nachtergaele. Bruno. 2001. Applied Analysis. World Scientific. 981-02-4191-7.
7. Book: Bachmann. George. Narici. Lawrence. Beckenstein. Edward. 2012-12-06. Fourier and Wavelet Analysis. Springer Science & Business Media. 9781461205050. 14.
8. Book: Hassani, Sadri. 1999. Mathematical Physics: A Modern Introduction to Its Foundations. Springer. 0-387-98579-4. 29. Equality holds iff =0 or c>=0. From the definition of c>, we conclude that a> and b> must be proportional..
9. Book: Axler. Sheldon. 2015. Linear Algebra Done Right, 3rd Ed.. Springer International Publishing. 978-3-319-11079-0. 172. This inequality is an equality if and only if one of u, v is a scalar multiple of the other..
10. Book: Bachman. George. Narici. Lawrence. 2012-09-26. Functional Analysis. Courier Corporation. 9780486136554. 141.
11. Book: Swartz, Charles. 1994-02-21. Measure, Integration and Function Spaces. World Scientific. 9789814502511. 236.
12. Book: Ricardo, Henry. 2009-10-21. A Modern Introduction to Linear Algebra. CRC Press. 9781439894613. 18.
13. Book: Banerjee. Sudipto. Roy. Anindya. 2014-06-06. Linear Algebra and Matrix Analysis for Statistics. CRC Press. 9781482248241. 181.
14. Book: Valenza, Robert J.. 2012-12-06. Linear Algebra: An Introduction to Abstract Mathematics. Springer Science & Business Media. 9781461209010. 146.
15. Book: Constantin, Adrian. 2016-05-21. Fourier Analysis with Applications. Cambridge University Press. 9781107044104. 74.
16. Book: Mukhopadhyay, Nitis. 2000-03-22. Probability and Statistical Inference. CRC Press. 9780824703790. 150.
17. Book: Keener, Robert W.. 2010-09-08. Theoretical Statistics: Topics for a Core Course. Springer Science & Business Media. 9780387938394. 71.
18. Wu. Hui-Hua. Wu. Shanhe. April 2009. Various proofs of the Cauchy-Schwarz inequality. Octogon Mathematical Magazine. 1222-5657. 978-973-88255-5-0. 17. 1. 221–229. https://ghostarchive.org/archive/20221009/http://www.uni-miskolc.hu/~matsefi/Octogon/volumes/volume1/article1_19.pdf . 2022-10-09 . live. 18 May 2016.
19. Book: Aliprantis. Charalambos D.. Border. Kim C.. 2007-05-02. Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer Science & Business Media. 9783540326960.
20. Book: Rudin, Walter. Real and Complex Analysis. McGraw-Hill. 1987. 0070542341. 3rd. New York. 1966.
21. Book: Faria. Edson de. Melo. Welington de. 2010-08-12. Mathematical Aspects of Quantum Field Theory. Cambridge University Press. 9781139489805. 273.
22. Book: Callebaut's inequality. Entry in the AoPS Wiki.
23. Moslehian. M.S.. Matharu. J.S.. Aujla. J.S.. 2011. Non-commutative Callebaut inequality. Linear Algebra and Its Applications. 436. 9. 3347–3353. 10.1016/j.laa.2011.11.024. 1112.3003. 119592971.
24. Liu, Shuangzhe. Neudecker, Heinz . 1999. A survey of Cauchy-Schwarz and Kantorovich-type matrix inequalities. Statistical Papers . 40 . 55–73 . 10.1007/BF02927110 . 122719088 .
25. Liu. Shuangzhe. Trenkler. Götz. Kollo. Tõnu. von Rosen. Dietrich. Baksalary. Oskar Maria. 2023. Professor Heinz Neudecker and matrix differential calculus. Statistical Papers. 65 . 4 . 2605–2639 . en. 10.1007/s00362-023-01499-w. 263661094 .