Cauchy-Schwarz Inequality [PDF]

Zitiervorschau

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] 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 inequality Proofs First proof Second proof More proofs Special cases Titu's lemma R2 (ordinary two-dimensional space) Rn (n-dimensional Euclidean space) L2 Applications Analysis Geometry Probability theory Generalizations See also Notes References External links

Statement of the inequality The Cauchy–Schwarz inequality states that for all vectors

where

and of an inner product space it is true that

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):

or

Proofs First proof 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

true. Similarly if

and

are also linearly dependent, regardless of , so the theorem is

. One henceforth assumes 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

apply the Pythagorean theorem to

which gives

(Indeed,

is the projection of

onto the plane orthogonal to

.) We can thus

and, after multiplication by

and taking square root, we get the Cauchy–Schwarz inequality. Moreover, if the relation

the above expression is actually an equality, then linear dependence between (since

and hence

and . On the other hand, if

and

; the definition of

in

then establishes a relation of

are linearly dependent, then there exists

such that

). Then

This establishes the theorem.

Second proof Let

and be arbitrary vectors in an inner product space over

In the special case

Therefore,

the theorem is trivially true. Now assume that

, or

If the inequality holds as an equality, then other hand, if

. . Let

be given by

, then

. , and so

and are linearly dependent, then

, thus

and

are linearly dependent. On the

, 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) 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.

L2 For the inner product space of square-integrable complex-valued functions, one has

A generalization of this is the Hölder inequality.

Applications Analysis 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.[8][9]

Geometry 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.

Generalizations 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 measure, the standard inner product gives rise to a positive functional functional

can be used to define an inner product

. Conversely, every positive linear , where

is the pointwise complex conjugate

Theorem (Cauchy–Schwarz inequality for positive functionals on C*-algebras):[17][18] If

is a positive linear functional on a

of

on

by

being a finite

. In this language, the Cauchy–Schwarz inequality becomes[16]

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

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 normal element

in its domain, we have

This extends the fact

and , when

is a unital positive map, then for every .

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

Bessel's inequality Jensen's inequality Kunita–Watanabe inequality Minkowski inequality

Notes 1. Steele, J. Michael (2004). The Cauchy–Schwarz Master Class: an Introduction to the Art of Mathematical Inequalities (http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/CSMC_index.html). 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. Hunter, John K.; Nachtergaele, Bruno (2001). Applied Analysis (https://books.google.com/books?id=oOYQVeHm Nk4C). World Scientific. ISBN 981-02-4191-7. 4. Bachmann, George; Narici, Lawrence; Beckenstein, Edward (2012-12-06). Fourier and Wavelet Analysis (https:// books.google.com/books?id=PkHhBwAAQBAJ). Springer Science & Business Media. p. 14. ISBN 9781461205050. 5. Hassani, Sadri (1999). Mathematical Physics: A Modern Introduction to Its Foundations. Springer. p. 29. ISBN 0387-98579-4. "Equality holds iff =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" (http://www.uni-miskol c.hu/~matsefi/Octogon/volumes/volume1/article1_19.pdf) (PDF). Octogon Mathematical Magazine. 17 (1): 221– 229. ISBN 978-973-88255-5-0. ISSN 1222-5657 (https://www.worldcat.org/issn/1222-5657). Retrieved 18 May 2016. 7. Aliprantis, Charalambos D.; Border, Kim C. (2007-05-02). Infinite Dimensional Analysis: A Hitchhiker's Guide (htt ps://books.google.com/books?id=4hIq6ExH7NoC). Springer Science & Business Media. ISBN 9783540326960. 8. Bachman, George; Narici, Lawrence (2012-09-26). Functional Analysis (https://books.google.com/books?id=_lTD AgAAQBAJ). Courier Corporation. p. 141. ISBN 9780486136554. 9. Swartz, Charles (1994-02-21). Measure, Integration and Function Spaces (https://books.google.com/books?id=S sbsCgAAQBAJ). World Scientific. p. 236. ISBN 9789814502511. 10. Ricardo, Henry (2009-10-21). A Modern Introduction to Linear Algebra (https://books.google.com/books?id=s7bM BQAAQBAJ). CRC Press. p. 18. ISBN 9781439894613. 11. Banerjee, Sudipto; Roy, Anindya (2014-06-06). Linear Algebra and Matrix Analysis for Statistics (https://books.go ogle.com/books?id=WDTcBQAAQBAJ). CRC Press. p. 181. ISBN 9781482248241. 12. Valenza, Robert J. (2012-12-06). Linear Algebra: An Introduction to Abstract Mathematics (https://books.google.c om/books?id=7x8MCAAAQBAJ). Springer Science & Business Media. p. 146. ISBN 9781461209010. 13. Constantin, Adrian (2016-05-21). Fourier Analysis with Applications (https://books.google.com/books?id=JnMZD AAAQBAJ). Cambridge University Press. p. 74. ISBN 9781107044104. 14. Mukhopadhyay, Nitis (2000-03-22). Probability and Statistical Inference (https://books.google.com/books?id=TM SnGkr_DxwC). CRC Press. p. 150. ISBN 9780824703790. 15. Keener, Robert W. (2010-09-08). Theoretical Statistics: Topics for a Core Course (https://books.google.com/book s?id=aVJmcega44cC). Springer Science & Business Media. p. 71. ISBN 9780387938394. 16. Faria, Edson de; Melo, Welington de (2010-08-12). Mathematical Aspects of Quantum Field Theory (https://book s.google.com/books?id=u9M9PFLNpMMC). Cambridge University Press. p. 273. ISBN 9781139489805. 17. Lin, Huaxin (2001-01-01). An Introduction to the Classification of Amenable C*-algebras (https://books.google.co m/books?id=2qru8d7BCAAC). World Scientific. p. 27. ISBN 9789812799883. 18. Arveson, W. (2012-12-06). An Invitation to C*-Algebras (https://books.google.com/books?id=d5TqBwAAQBAJ). Springer Science & Business Media. p. 28. ISBN 9781461263715.

19. Størmer, Erling (2012-12-13). Positive Linear Maps of Operator Algebras (https://books.google.com/books?id=lQt KAIONqwIC). 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 (https://doi.org/10.2307%2F1969657). JSTOR 1969657 (https://www.jstor.org/stable/1969657). 21. Paulsen, Vern (2002). Completely Bounded Maps and Operator Algebras (https://books.google.com/books?id=Vt SFHDABxMIC&pg=PA40). 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 (https://doi.org/10.1016%2F0022-247X%2865%2990016-8). 23. Callebaut's inequality (https://artofproblemsolving.com/wiki/index.php?title=Callebaut%27s_Inequality). Entry in the AoPS Wiki. 24. Moslehian, M.S.; Matharu, J.S.; Aujla, J.S. (2011). "Non-commutative Callebaut inequality". arXiv:1112.3003 (htt ps://arxiv.org/abs/1112.3003) [math.FA (https://arxiv.org/archive/math.FA)].

References Aldaz, J. M.; Barza, S.; Fujii, M.; Moslehian, M. S. (2015), "Advances in Operator Cauchy—Schwarz inequalities and their reverses", Annals of Functional Analysis, 6 (3): 275–295, doi:10.15352/afa/06-3-20 (https://doi.org/10.1 5352%2Fafa%2F06-3-20) Bityutskov, V. I. (2001) [1994], "Bunyakovskii inequality" (https://www.encyclopediaofmath.org/index.php?title=b/b 017770), in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 Bunyakovsky, V. (1859), "Sur quelques inegalités concernant les intégrales aux différences finies" (http://www-sta t.wharton.upenn.edu/~steele/Publications/Books/CSMC/bunyakovsky.pdf) (PDF), Mem. Acad. Sci. St. Petersbourg, 7 (1): 9 Cauchy, A.-L. (1821), "Sur les formules qui résultent de l'emploie du signe et sur > ou