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Cauchy–Schwarz 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] It has a number of generalizations, among them Hölder's inequality.

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

Contents

Statement of the inequalityEdit

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]:14

If   and   have an imaginary component, the inner product is the standard complex inner product, where the bar notation is used for complex conjugation, and then the inequality may be restated more explicitly as

 

or

 

ProofsEdit

First proofEdit

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 we have equality, and in this case   and   are also linearly dependent, regardless of  , so the theorem is true. Similarly if  . We henceforth assume that   is nonzero.

Let

 

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.

Second proofEdit

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

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 proofsEdit

There are indeed many different proofs[5] 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  .[6]

Special casesEdit

R2 (ordinary two-dimensional space)Edit

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)Edit

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.

L2Edit

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

 

A generalization of this is the Hölder inequality.

ApplicationsEdit

AnalysisEdit

The triangle inequality for the standard 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.[7][8]

GeometryEdit

The Cauchy–Schwarz inequality allows one to extend the notion of "angle between two vectors" to any real inner-product space by defining:[9][10]

 

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,[11][12] as is done when extracting a metric from quantum fidelity.

Probability theoryEdit

Let X, Y be random variables, then the covariance inequality[13][14] is given by

 

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

 

then the Cauchy–Schwarz inequality becomes

 

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

 

where   denotes variance, and   denotes covariance.

GeneralizationsEdit

Various generalizations of the Cauchy–Schwarz inequality exist 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[15]

 

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

Theorem (Cauchy–Schwarz inequality for positive functionals on C*-algebras):[16][17] 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,[18][19] 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):[20] For a 2-positive map   between C*-algebras, for all   in its domain,

 
 

See alsoEdit

NotesEdit

  1. ^ a b c Steele, J. Michael. "The Cauchy–Schwarz Master Class: an Introduction to the Art of Mathematical Inequalities, Ch. 1". 
  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 (2001-01-01). Applied Analysis. World Scientific. ISBN 9789810241919. 
  4. ^ Bachmann, George; Narici, Lawrence; Beckenstein, Edward (2012-12-06). Fourier and Wavelet Analysis. Springer Science & Business Media. ISBN 9781461205050. 
  5. ^ 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 18 May 2016. 
  6. ^ Aliprantis, Charalambos D.; Border, Kim C. (2007-05-02). Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer Science & Business Media. ISBN 9783540326960. 
  7. ^ Bachman, George; Narici, Lawrence (2012-09-26). Functional Analysis. Courier Corporation. p. 141. ISBN 9780486136554. 
  8. ^ Swartz, Charles (1994-02-21). Measure, Integration and Function Spaces. World Scientific. p. 236. ISBN 9789814502511. 
  9. ^ Ricardo, Henry (2009-10-21). A Modern Introduction to Linear Algebra. CRC Press. p. 18. ISBN 9781439894613. 
  10. ^ Banerjee, Sudipto; Roy, Anindya (2014-06-06). Linear Algebra and Matrix Analysis for Statistics. CRC Press. p. 181. ISBN 9781482248241. 
  11. ^ Valenza, Robert J. (2012-12-06). Linear Algebra: An Introduction to Abstract Mathematics. Springer Science & Business Media. p. 146. ISBN 9781461209010. 
  12. ^ Constantin, Adrian (2016-05-21). Fourier Analysis with Applications. Cambridge University Press. p. 74. ISBN 9781107044104. 
  13. ^ Mukhopadhyay, Nitis (2000-03-22). Probability and Statistical Inference. CRC Press. p. 150. ISBN 9780824703790. 
  14. ^ Keener, Robert W. (2010-09-08). Theoretical Statistics: Topics for a Core Course. Springer Science & Business Media. p. 71. ISBN 9780387938394. 
  15. ^ Faria, Edson de; Melo, Welington de (2010-08-12). Mathematical Aspects of Quantum Field Theory. Cambridge University Press. p. 273. ISBN 9781139489805. 
  16. ^ Lin, Huaxin (2001-01-01). An Introduction to the Classification of Amenable C*-algebras. World Scientific. p. 27. ISBN 9789812799883. 
  17. ^ Arveson, W. (2012-12-06). An Invitation to C*-Algebras. Springer Science & Business Media. p. 28. ISBN 9781461263715. 
  18. ^ Størmer, Erling (2012-12-13). Positive Linear Maps of Operator Algebras. Springer Science & Business Media. ISBN 9783642343698. 
  19. ^ 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. 
  20. ^ Paulsen (2002), Completely Bounded Maps and Operator Algebras, ISBN 9780521816694  page 40.

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