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,
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:
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.
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 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.
The Cauchy–Schwarz inequality is usually used to show Bessel’s inequality.
If either or are the zero vector, the statement holds trivially, so assume that both are nonzero.
For any nonzero vector , (NOTE: merits own proof)
If the inner product is symmetric. Let be a real scalar.
The last expression is a quadratic polynomial that is non-negative for any . The quadratic has either two complex roots,or a single real root. Intuitively, the polynomial is either ‘floating above’ the horizontal axis, if it has two complex roots, or tangent to it if it has one real root, since it can’t have two real roots because the graph of the function would have to ‘pass under’ the horizontal axis and take some negative values.
The roots are given by the quadratic formula
If the inner product is symmetric, this proves the inequality.
An alternative proof follows from the expression
valid for and real and $x>0$ and $y>0$. This expression is a restatement of . From this one can get a general n-term expression
To get cauchy-Scwarz inequality set and .
If the inner product is skew-symmetric, take
For a real convex function , numbers x1, x2, …, xn in its domain, and positive weights ai, Jensen’s inequality can be stated as:
and the inequality is reversed if is concave, which is
- The arithmetic mean, or less precisely the average, of a list of n numbers x1, x2, . . ., xn is the sum of the numbers divided by n:
The geometric mean is similar, except that it is only defined for a list of nonnegative real numbers, and uses multiplication and a root in place of addition and division:
For any list of n nonnegative real numbers x1, x2, . . ., xn,
and that equality holds if and only if x1 = x2 = . . . = xn.
Equality holds if and only if ap = bq. This form of Young’s inequality is a special case of the inequality of weighted arithmetic and geometric means and can be used to prove Hölder’s inequality.
with equality if and only if ap = bq. Young’s inequality follows by exponentiating.