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In mathematics, the Cauchy–Schwarz inequality (also known as the Bunyakovsky inequality, the Schwarz inequality, or theCauchy–Bunyakovsky–Schwarz inequality), is a useful inequality encountered in many different settings, such as linear algebra,analysis, in probability theory, and other areas. It is a specific case of Hölder’s inequality.
The inequality for sums was published by Augustin-Louis Cauchy (1821), while the corresponding inequality for integrals was first stated by Viktor Bunyakovsky (1859) and rediscovered by Hermann Amandus Schwarz (1888) (often misspelled “Schwartz”).
Statement of the inequality
The Cauchy–Schwarz inequality states that for all vectors x and y of an inner product space,
where is the inner product. Equivalently, by taking the square root of both sides, and referring to the norms of the vectors, the inequality is written as
Moreover, the two sides are equal if and only if x and y are linearly dependent (or, in a geometrical sense, they are parallel or one of the vectors is equal to zero).
If and are any complex numbers and the inner product is the standard inner product then the inequality may be restated in a more explicit way as follows:
When viewed in this way the numbers x1, …, xn, and y1, …, yn are the components of x and y with respect to an orthonormal basis of V.
Even more compactly written:
Equality holds if and only if x and y are linearly dependent, that is, one is a scalar multiple of the other (which includes the case when one or both are zero).
The finite-dimensional case of this inequality for real vectors was proved by Cauchy in 1821, and in 1859 Cauchy’s studentBunyakovsky noted that by taking limits one can obtain an integral form of Cauchy’s inequality. The general result for an inner product space was obtained by Schwarz in 1885.
Let u, v be arbitrary vectors in a vector space V over F with an inner product, where F is the field of real or complex numbers. We prove the inequality
This inequality is trivial in the case v = 0, so we assume that <v, v> is nonzero. Let δ be any number in the field F. Then,
Choose the value of δ that minimizes this quadratic form, namely
(A quick way to remember this value of δ is to imagine F to be the reals, so that the quadratic form is a quadratic polynomial in the real variable δ, and the polynomial can easily be minimized by setting its derivative equal to zero.)
which is true if and only if
which completes the proof.
Notable special cases
In Euclidean space Rn with the standard inner product, the Cauchy–Schwarz inequality is
To prove this form of the inequality, consider the following quadratic polynomial in z.
Since it is nonnegative it has at most one real root in z, whence its discriminant is less than or equal to zero, that is,
which yields the Cauchy–Schwarz inequality.
An equivalent proof for Rn starts with the summation below.
Expanding the brackets we have:
collecting together identical terms (albeit with different summation indices) we find:
Because the left-hand side of the equation is a sum of the squares of real numbers it is greater than or equal to zero, thus:
This form is used usually when solving school math problems.
Yet another approach when n ≥ 2 (n = 1 is trivial) is to consider the plane containing x and y. More precisely, recoordinatize Rn with any orthonormal basis whose first two vectors span a subspace containing x and y. In this basis only and are nonzero, and the inequality reduces to the algebra of dot product in the plane, which is related to the angle between two vectors, from which we obtain the inequality:
When n = 3 the Cauchy–Schwarz inequality can also be deduced from Lagrange’s identity, which takes the form
from which readily follows 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 inner product 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 allows one to extend the notion of “angle between two vectors” to any real inner product space, by defining:
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 of the right hand side, as is done when extracting a metric from quantum fidelity.
The Cauchy–Schwarz is used to prove that the inner product is a continuous function with respect to the topology induced by the inner product itself.
The Cauchy–Schwarz inequality is usually used to show Bessel’s inequality.
If the inner product is symmetric. Let be a real scalar.
If the inner product is symmetric, this proves the inequality.