Cauchy-Schwarz Inequality
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Cauchy%E2%80%93Schwarz Inequality

In mathematics, the Cauchy-Schwarz inequality, also known as the Cauchy-Bunyakovsky-Schwarz inequality, is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, vector algebra and other areas. It is considered to be one of the most important inequalities in all of mathematics.[1]

The inequality for sums was published by Augustin-Louis Cauchy (1821), while the corresponding inequality for integrals was first proved by Viktor Bunyakovsky (1859). Later the integral inequality was rediscovered by Hermann Amandus Schwarz (1888).[1]

Statement of the inequality

The Cauchy-Schwarz inequality states that for all vectors and of an inner product space it is true that

where is the inner product. Examples of inner products include the real and complex dot product; see the examples in inner product. Equivalently, by taking the square root of both sides, and referring to the norms of the vectors, the inequality is written as[2][3]

Moreover, the two sides are equal if and only if and are linearly dependent (meaning they are parallel: one of the vector's magnitudes is zero, or one is a scalar multiple of the other).[4][5]

If and , and the inner product is the standard complex inner product, then the inequality may be restated more explicitly as follows (where the bar notation is used for complex conjugation): for , we have

That is,


Proof 1 —

Let and be arbitrary vectors in a vector space over with an inner product, where is the field of real or complex numbers. We prove the inequality

and that equality holds if and only if either or is a multiple of the other (which includes the special case that either is the zero vector).

If , it is clear that there is equality, and in this case and are also linearly dependent, regardless of , so the theorem is true. Similarly if . One henceforth assumes that is nonzero.


Then, by linearity of the inner product in its first argument, one has

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

and, after multiplication by and taking square root, we get the Cauchy-Schwarz inequality. 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 . On the other hand, if and are linearly dependent, then there exists such that (since ). Then

This establishes the theorem.

Proof 2 —

Let and be arbitrary vectors in an inner product space over .

In the special case the theorem is trivially true. Now assume that . Let be given by , then

Therefore, , or .

If the inequality holds as an equality, then , and so , thus and are linearly dependent. On the other hand, if and are linearly dependent, then , as shown in the first proof.

More proofs

There are many different proofs[6] of the Cauchy-Schwarz inequality other than the above two examples.[1][3] 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 .[7]

Special cases

Titu's lemma

Titu's lemma (named after Titu Andreescu, also known as T2 lemma, Engel's form, or Sedrakyan's inequality) states that for positive reals, one has

It is a direct consequence of the Cauchy-Schwarz inequality, obtained upon substituting and This form is especially helpful when the inequality involves fractions where the numerator is a perfect square.

R2 (ordinary two-dimensional space)

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

In the usual 2-dimensional space with the dot product, let and . The Cauchy-Schwarz inequality is that

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)

In Euclidean space with the standard inner product, the Cauchy-Schwarz inequality is

The Cauchy-Schwarz inequality can be proved using only ideas from elementary algebra in this case. Consider the following quadratic polynomial in

Since it is nonnegative, it has at most one real root for , hence its discriminant is less than or equal to zero. That is,

which yields the Cauchy-Schwarz inequality.


For the inner product space of square-integrable complex-valued functions, one has

A generalization of this is the Hölder inequality.



The triangle inequality for the Euclidean norm is often shown as a consequence of the Cauchy-Schwarz inequality, as follows.

Given vectors x and y,

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.[8][9]


The Cauchy-Schwarz inequality allows one to extend the notion of "angle between two vectors" to any real inner-product space by defining:[10][11]

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,[12][13] as is done when extracting a metric from quantum fidelity.

Probability theory

Let X, Y be random variables, then the covariance inequality[14][15] 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.


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[16]

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

Theorem (Cauchy-Schwarz inequality for positive functionals on C*-algebras):[17][18] If is a positive linear functional on a C*-algebra then for all , .

The next two theorems are further examples in operator algebra.

Theorem (Kadison-Schwarz inequality,[19][20] 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, i.e. is sometimes known as Kadison's inequality.

Theorem (Modified Schwarz inequality for 2-positive maps):[21] For a 2-positive map between C*-algebras, for all in its domain,

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

Theorem (Callebaut's Inequality)[22] For reals ,

It can be easily proven by Hölder's inequality.[23] There are also non commutative versions for operators and tensor products of matrices.[24]

See also


  1. ^ 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.
  2. ^ Strang, Gilbert (19 July 2005). "3.2". Linear Algebra and its Applications (4th ed.). Stamford, CT: Cengage Learning. pp. 154-155. ISBN 978-0030105678.
  3. ^ a b Hunter, John K.; Nachtergaele, Bruno (2001). Applied Analysis. World Scientific. ISBN 981-02-4191-7.
  4. ^ Bachmann, George; Narici, Lawrence; Beckenstein, Edward (2012-12-06). Fourier and Wavelet Analysis. Springer Science & Business Media. p. 14. ISBN 9781461205050.
  5. ^ 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.
  6. ^ 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. Retrieved 2016.
  7. ^ Aliprantis, Charalambos D.; Border, Kim C. (2007-05-02). Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer Science & Business Media. ISBN 9783540326960.
  8. ^ Bachman, George; Narici, Lawrence (2012-09-26). Functional Analysis. Courier Corporation. p. 141. ISBN 9780486136554.
  9. ^ Swartz, Charles (1994-02-21). Measure, Integration and Function Spaces. World Scientific. p. 236. ISBN 9789814502511.
  10. ^ Ricardo, Henry (2009-10-21). A Modern Introduction to Linear Algebra. CRC Press. p. 18. ISBN 9781439894613.
  11. ^ Banerjee, Sudipto; Roy, Anindya (2014-06-06). Linear Algebra and Matrix Analysis for Statistics. CRC Press. p. 181. ISBN 9781482248241.
  12. ^ Valenza, Robert J. (2012-12-06). Linear Algebra: An Introduction to Abstract Mathematics. Springer Science & Business Media. p. 146. ISBN 9781461209010.
  13. ^ Constantin, Adrian (2016-05-21). Fourier Analysis with Applications. Cambridge University Press. p. 74. ISBN 9781107044104.
  14. ^ Mukhopadhyay, Nitis (2000-03-22). Probability and Statistical Inference. CRC Press. p. 150. ISBN 9780824703790.
  15. ^ Keener, Robert W. (2010-09-08). Theoretical Statistics: Topics for a Core Course. Springer Science & Business Media. p. 71. ISBN 9780387938394.
  16. ^ Faria, Edson de; Melo, Welington de (2010-08-12). Mathematical Aspects of Quantum Field Theory. Cambridge University Press. p. 273. ISBN 9781139489805.
  17. ^ Lin, Huaxin (2001-01-01). An Introduction to the Classification of Amenable C*-algebras. World Scientific. p. 27. ISBN 9789812799883.
  18. ^ Arveson, W. (2012-12-06). An Invitation to C*-Algebras. Springer Science & Business Media. p. 28. ISBN 9781461263715.
  19. ^ Størmer, Erling (2012-12-13). Positive Linear Maps of Operator Algebras. Springer Monographs in Mathematics. Springer Science & Business Media. ISBN 9783642343698.
  20. ^ 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.
  21. ^ Paulsen, Vern (2002). Completely Bounded Maps and Operator Algebras. Cambridge Studies in Advanced Mathematics. 78. Cambridge University Press. p. 40. ISBN 9780521816694.
  22. ^ 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.
  23. ^ Callebaut's inequality. Entry in the AoPS Wiki.
  24. ^ Moslehian, M.S.; Matharu, J.S.; Aujla, J.S. (2011). "Non-commutative Callebaut inequality". arXiv:1112.3003 [math.FA].


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