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Cauchy–Schwarz inequality

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The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality)[1][2][3][4] 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.[5]

The inequality for sums was published by Augustin-Louis Cauchy (1821). The corresponding inequality for integrals was published by Viktor Bunyakovsky (1859)[2] and Hermann Schwarz (1888). Schwarz gave the modern proof of the integral version.[5]

Statement of the inequality[edit]

The Cauchy–Schwarz inequality states that for all vectors and of an inner product space


where 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 norm, called the canonical or induced norm, where the norm of a vector is denoted and defined by where 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:[6][7]


Moreover, the two sides are equal if and only if and are linearly dependent.[8][9][10]

Special cases[edit]

Sedrakyan's lemma - Positive real numbers[edit]

Sedrakyan's inequality, also known as Bergström's inequality, Engel's form, Titu's lemma (or the T2 lemma), states that for real numbers and positive real numbers :

It is a direct consequence of the Cauchy–Schwarz inequality, obtained by using the dot product on upon substituting and . This form is especially helpful when the inequality involves fractions where the numerator is a perfect square.

R2 - The plane[edit]

Cauchy-Schwarz inequality in a unit circle of the Euclidean plane

The real vector space denotes the 2-dimensional plane. It is also the 2-dimensional Euclidean space where the inner product is the dot product. If and then the Cauchy–Schwarz inequality becomes: where is the angle between and .

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 , , , and as where equality holds if and only if the vector is in the same or opposite direction as the vector , or if one of them is the zero vector.

Rn: n-dimensional Euclidean space[edit]

In Euclidean space with the standard inner product, which is the dot product, the Cauchy–Schwarz inequality becomes:

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

or by considering the following quadratic polynomial in

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,

Cn: n-dimensional Complex space[edit]

If with and (where and ) and if the inner product on the vector space is the canonical complex inner product (defined by where the bar notation is used for complex conjugation), then the inequality may be restated more explicitly as follows:

That is,


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

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:

Taking square roots gives the triangle inequality:

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.[11][12]


The Cauchy–Schwarz inequality allows one to extend the notion of "angle between two vectors" to any real inner-product space by defining:[13][14]

The Cauchy–Schwarz inequality proves that this definition is sensible, by showing that the right-hand side lies in the interval [−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,[15][16] as is done when extracting a metric from quantum fidelity.

Probability theory[edit]

Let and be random variables. Then the covariance inequality[17][18] is given by:

After defining an inner product on the set of random variables using the expectation of their product, the Cauchy–Schwarz inequality becomes

To prove the covariance inequality using the Cauchy–Schwarz inequality, let and then where denotes variance and denotes covariance.


There are many different proofs[19] of the Cauchy–Schwarz inequality other than those given below.[5][7] 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 and not [20]

This section gives two proofs of the following theorem:

Cauchy–Schwarz inequality — Let and be arbitrary vectors in an inner product space over the scalar field where is the field of real numbers or complex numbers Then

(Cauchy–Schwarz Inequality)

with equality holding in the Cauchy–Schwarz Inequality if and only if and are linearly dependent.

Moreover, if and then

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 ) 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 and are linearly dependent then

Proof of the trivial parts: Case where a vector is and also one direction of the Equality Characterization

By definition, and are linearly dependent if and only if one is a scalar multiple of the other. If where is some scalar then

which shows that equality holds in the Cauchy–Schwarz Inequality. The case where for some scalar follows from the previous case:

In particular, if at least one of and is the zero vector then and are necessarily linearly dependent (for example, if then where ), 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[edit]

The special case of was proven above so it is henceforth assumed that Let

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

Therefore, is a vector orthogonal to the vector (Indeed, is the projection of onto the plane orthogonal to ) We can thus apply the Pythagorean theorem to which gives

The Cauchy–Schwarz inequality follows by multiplying by and then taking the square root. Moreover, if the relation in the above expression is actually an equality, then and hence the definition of then establishes a relation of linear dependence between and The converse was proved at the beginning of this section, so the proof is complete.

Proof by analyzing a quadratic[edit]

Consider an arbitrary pair of vectors . Define the function defined by , where is a complex number satisfying and . Such an exists since if then can be taken to be 1.

Since the inner product is positive-definite, only takes non-negative real values. On the other hand, can be expanded using the bilinearity of the inner product: Thus, is a polynomial of degree (unless which is a case that was checked earlier). Since the sign of does not change, the discriminant of this polynomial must be non-positive: The conclusion follows.[21]

For the equality case, notice that happens if and only if If then and hence


Various generalizations of the Cauchy–Schwarz inequality exist. Hölder's inequality generalizes it to 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 being a finite measure, the standard inner product gives rise to a positive functional by Conversely, every positive linear functional on can be used to define an inner product where is the pointwise complex conjugate of In this language, the Cauchy–Schwarz inequality becomes[22]

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

Cauchy–Schwarz inequality for positive functionals on C*-algebras[23][24] — If is a positive linear functional on a C*-algebra then for all

The next two theorems are further examples in operator algebra.

Kadison–Schwarz inequality[25][26] (Named after Richard Kadison) — If is a unital positive map, then for every normal element in its domain, we have and

This extends the fact when is a linear functional. The case when is self-adjoint, that is, is sometimes known as Kadison's inequality.

Cauchy–Schwarz inequality (Modified Schwarz inequality for 2-positive maps[27]) — For a 2-positive map between C*-algebras, for all in its domain,

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

Callebaut's Inequality[28] — For reals

This theorem can be deduced from Hölder's inequality.[29] There are also non-commutative versions for operators and tensor products of matrices.[30]

Several matrix versions of the Cauchy–Schwarz inequality and Kantorovich inequality are applied to linear regression models.[31] [32]

See also[edit]



  1. ^ O'Connor, J.J.; Robertson, E.F. "Hermann Amandus Schwarz". University of St Andrews, Scotland.
  2. ^ a b Bityutskov, V. I. (2001) [1994], "Bunyakovskii inequality", Encyclopedia of Mathematics, EMS Press
  3. ^ Ćurgus, Branko. "Cauchy-Bunyakovsky-Schwarz inequality". Department of Mathematics. Western Washington University.
  4. ^ Joyce, David E. "Cauchy's inequality" (PDF). Department of Mathematics and Computer Science. Clark University. Archived (PDF) from the original on 2022-10-09.
  5. ^ a b c Steele, J. Michael (2004). The Cauchy–Schwarz Master Class: an Introduction to the Art of Mathematical Inequalities. The Mathematical Association of America. p. 1. ISBN 978-0521546775. ...there is no doubt that this is one of the most widely used and most important inequalities in all of mathematics.
  6. ^ Strang, Gilbert (19 July 2005). "3.2". Linear Algebra and its Applications (4th ed.). Stamford, CT: Cengage Learning. pp. 154–155. ISBN 978-0030105678.
  7. ^ a b Hunter, John K.; Nachtergaele, Bruno (2001). Applied Analysis. World Scientific. ISBN 981-02-4191-7.
  8. ^ Bachmann, George; Narici, Lawrence; Beckenstein, Edward (2012-12-06). Fourier and Wavelet Analysis. Springer Science & Business Media. p. 14. ISBN 9781461205050.
  9. ^ Hassani, Sadri (1999). Mathematical Physics: A Modern Introduction to Its Foundations. Springer. p. 29. ISBN 0-387-98579-4. Equality holds iff <c|c>=0 or |c>=0. From the definition of |c>, we conclude that |a> and |b> must be proportional.
  10. ^ Axler, Sheldon (2015). Linear Algebra Done Right, 3rd Ed. Springer International Publishing. p. 172. ISBN 978-3-319-11079-0. This inequality is an equality if and only if one of u, v is a scalar multiple of the other.
  11. ^ Bachman, George; Narici, Lawrence (2012-09-26). Functional Analysis. Courier Corporation. p. 141. ISBN 9780486136554.
  12. ^ Swartz, Charles (1994-02-21). Measure, Integration and Function Spaces. World Scientific. p. 236. ISBN 9789814502511.
  13. ^ Ricardo, Henry (2009-10-21). A Modern Introduction to Linear Algebra. CRC Press. p. 18. ISBN 9781439894613.
  14. ^ Banerjee, Sudipto; Roy, Anindya (2014-06-06). Linear Algebra and Matrix Analysis for Statistics. CRC Press. p. 181. ISBN 9781482248241.
  15. ^ Valenza, Robert J. (2012-12-06). Linear Algebra: An Introduction to Abstract Mathematics. Springer Science & Business Media. p. 146. ISBN 9781461209010.
  16. ^ Constantin, Adrian (2016-05-21). Fourier Analysis with Applications. Cambridge University Press. p. 74. ISBN 9781107044104.
  17. ^ Mukhopadhyay, Nitis (2000-03-22). Probability and Statistical Inference. CRC Press. p. 150. ISBN 9780824703790.
  18. ^ Keener, Robert W. (2010-09-08). Theoretical Statistics: Topics for a Core Course. Springer Science & Business Media. p. 71. ISBN 9780387938394.
  19. ^ Wu, Hui-Hua; Wu, Shanhe (April 2009). "Various proofs of the Cauchy-Schwarz inequality" (PDF). Octogon Mathematical Magazine. 17 (1): 221–229. ISBN 978-973-88255-5-0. ISSN 1222-5657. Archived (PDF) from the original on 2022-10-09. Retrieved 18 May 2016.
  20. ^ Aliprantis, Charalambos D.; Border, Kim C. (2007-05-02). Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer Science & Business Media. ISBN 9783540326960.
  21. ^ Rudin, Walter (1987) [1966]. Real and Complex Analysis (3rd ed.). New York: McGraw-Hill. ISBN 0070542341.
  22. ^ Faria, Edson de; Melo, Welington de (2010-08-12). Mathematical Aspects of Quantum Field Theory. Cambridge University Press. p. 273. ISBN 9781139489805.
  23. ^ Lin, Huaxin (2001-01-01). An Introduction to the Classification of Amenable C*-algebras. World Scientific. p. 27. ISBN 9789812799883.
  24. ^ Arveson, W. (2012-12-06). An Invitation to C*-Algebras. Springer Science & Business Media. p. 28. ISBN 9781461263715.
  25. ^ Størmer, Erling (2012-12-13). Positive Linear Maps of Operator Algebras. Springer Monographs in Mathematics. Springer Science & Business Media. ISBN 9783642343698.
  26. ^ Kadison, Richard V. (1952-01-01). "A Generalized Schwarz Inequality and Algebraic Invariants for Operator Algebras". Annals of Mathematics. 56 (3): 494–503. doi:10.2307/1969657. JSTOR 1969657.
  27. ^ Paulsen, Vern (2002). Completely Bounded Maps and Operator Algebras. Cambridge Studies in Advanced Mathematics. Vol. 78. Cambridge University Press. p. 40. ISBN 9780521816694.
  28. ^ Callebaut, D.K. (1965). "Generalization of the Cauchy–Schwarz inequality". J. Math. Anal. Appl. 12 (3): 491–494. doi:10.1016/0022-247X(65)90016-8.
  29. ^ Callebaut's inequality. Entry in the AoPS Wiki.
  30. ^ Moslehian, M.S.; Matharu, J.S.; Aujla, J.S. (2011). "Non-commutative Callebaut inequality". Linear Algebra and Its Applications. 436 (9): 3347–3353. arXiv:1112.3003. doi:10.1016/j.laa.2011.11.024. S2CID 119592971.
  31. ^ Liu, Shuangzhe; Neudecker, Heinz (1999). "A survey of Cauchy-Schwarz and Kantorovich-type matrix inequalities". Statistical Papers. 40: 55–73. doi:10.1007/BF02927110. S2CID 122719088.
  32. ^ Liu, Shuangzhe; Trenkler, Götz; Kollo, Tõnu; von Rosen, Dietrich; Baksalary, Oskar Maria (2023). "Professor Heinz Neudecker and matrix differential calculus". Statistical Papers. doi:10.1007/s00362-023-01499-w. S2CID 263661094.


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