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.
The inequality for sums was published by, while the corresponding inequality for integrals was first proved by
. Later the integral inequality was rediscovered by.

Statement of the inequality

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
Moreover, the two sides are equal if and only if and are linearly dependent.
If and, and the inner product is the standard complex inner product, then the inequality may be restated more explicitly as follows :
or

Proofs

;More proofs
There are many different proofs of the Cauchy–Schwarz inequality other than the above two examples.
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.

Special cases

Titu's lemma

Titu's lemma 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.

R''n'' (''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.

Geometry

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 and justifies the notion that 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, as is done when extracting a metric from quantum fidelity.

Probability theory

Let X, Y be random variables, then the covariance inequality 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. 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. 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
which extends verbatim to positive functionals on C*-algebras:
Theorem : If is a positive linear functional on a C*-algebra then for all,.
The next two theorems are further examples in operator algebra.
Theorem : 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 : 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
For reals,
It can be easily proven by Hölder's inequality. There are also non commutative versions for operators and tensor products of matrices.