Cauchy–Schwarz inequality

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The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky-Schwarz inequality)[1][2][3][4] 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 it is true that

 

 

 

 

(Cauchy-Schwarz inequality [written using only the inner product])

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

so that this norm and the inner product are related by the defining condition 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:[6][7]

 

 

 

 

(Cauchy-Schwarz inequality [written using norm and inner product])

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

Special cases[edit]

Titu's lemma - Positive real numbers[edit]

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.

2 - 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.

n - 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 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.

n - 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 ), then the inequality may be restated more explicitly as follows (where the bar notation is used for complex conjugation):

That is,

L2[edit]

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

The Hölder inequality is a generalization of this.

Proofs[edit]

There are many different proofs[11] 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 [12]

This section gives 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)

where in addition, equality holds in the Cauchy-Schwarz Inequality if and only if and are linearly dependent.

Moreover, if this equality holds and if then

In all 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 the above Equality Characterization; 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 is very similar, with the main difference between the complex conjugation of :

If at least one of and is the zero vector then and are necessarily linearly dependent (just scalar multiply the non-zero vector by the number to get the zero vector; e.g. if then let so that ), which proves the converse of this characterization in this special case; that is, this shows that if at least one of and is then the Equality Characterization holds.

If which happens if and only if then and so that in particular, the Cauchy-Schwarz inequality holds because both sides of it are The proof in the case of is identical.

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.

For real inner products spaces[edit]

Let be an real inner product space. Consider an arbitary pair and the function defined by Since the inner product is positive-definite, only takes non-negative values. On the other hand, can be expanded using the bilinearity of the inner product and using the fact that for real inner products:

Thus, is a polynomial of degree (unless which is a case that can be independently verified). Since the sign of does not change, the discriminant of this polynomial must be non-positive:
The conclusion follows.

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

For arbitrary vector spaces[edit]

Proof 1[edit]

The special case of was proven above so it is henceforth assumed that As is now shown, the Cauchy–Schwarz inequality (and the rest of the theorem) is an almost immediate corrollary of the following equality:

[13]

 

 

 

 

(Eq. 1)

which is readily verified by elementarily expanding (via the definition of the norm) and then simplifying.

Observing that the left hand side (LHS) of Eq. 1 is non-negative (which makes this also true of the right hand side (RHS)) proves that from which the Cauchy-Schwarz Inequality follows (by taking the square root of both sides). If then the RHS (and thus also the LHS) of Eq. 1 is which is only possible if ;[note 1] thus which shows that and are linearly dependent.[13] Since the (trivial) converse was proved above, the proof of the theorem is complete.


Details of 's elementary expansion are now given for the interested reader. Let and so that and Then

This expansion does not require to be non-zero; however, must be non-zero in order to divide both sides by and to deduce the Cauchy-Schwarz inequality from it. Swapping and gives rise to:

and thus

Proof 2[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.

Applications[edit]

Analysis[edit]

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.[14][15]

Geometry[edit]

The Cauchy–Schwarz inequality allows one to extend the notion of "angle between two vectors" to any real inner-product space by defining:[16][17]

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,[18][19] as is done when extracting a metric from quantum fidelity.

Probability theory[edit]

Let and be random variables, then the covariance inequality:[20][21] 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.

Generalizations[edit]

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, i.e. 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]

See also[edit]

Notes[edit]

  1. ^ In fact, it follows immediately from Eq. 1 that if and only if

Citations[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.
  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. ^ 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.
  12. ^ Aliprantis, Charalambos D.; Border, Kim C. (2007-05-02). Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer Science & Business Media. ISBN 9783540326960.
  13. ^ a b Halmos 1982, pp. 2, 167.
  14. ^ Bachman, George; Narici, Lawrence (2012-09-26). Functional Analysis. Courier Corporation. p. 141. ISBN 9780486136554.
  15. ^ Swartz, Charles (1994-02-21). Measure, Integration and Function Spaces. World Scientific. p. 236. ISBN 9789814502511.
  16. ^ Ricardo, Henry (2009-10-21). A Modern Introduction to Linear Algebra. CRC Press. p. 18. ISBN 9781439894613.
  17. ^ Banerjee, Sudipto; Roy, Anindya (2014-06-06). Linear Algebra and Matrix Analysis for Statistics. CRC Press. p. 181. ISBN 9781482248241.
  18. ^ Valenza, Robert J. (2012-12-06). Linear Algebra: An Introduction to Abstract Mathematics. Springer Science & Business Media. p. 146. ISBN 9781461209010.
  19. ^ Constantin, Adrian (2016-05-21). Fourier Analysis with Applications. Cambridge University Press. p. 74. ISBN 9781107044104.
  20. ^ Mukhopadhyay, Nitis (2000-03-22). Probability and Statistical Inference. CRC Press. p. 150. ISBN 9780824703790.
  21. ^ Keener, Robert W. (2010-09-08). Theoretical Statistics: Topics for a Core Course. Springer Science & Business Media. p. 71. ISBN 9780387938394.
  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. 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". arXiv:1112.3003 [math.FA].

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