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 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).
Statement of the inequalityEdit
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
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):
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 we have equality, and in this case and are also linearly dependent, regardless of , so the theorem is true. Similarly if . We henceforth assume that is nonzero.
Then, by linearity of the inner product in its first argument, one has
and, after multiplication by and taking square root, we get the Cauchy–Schwarz inequality. 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 . On the other hand, if and are linearly dependent, then there exists such that (since ). Then
This establishes the theorem.
Let and be arbitrary vectors in an inner product space over .
In the special case the theorem is trivially true. Now assume that . Let be given by , then
Therefore, , or .
If the inequality holds as an equality, then , and so , thus and are linearly dependent. On the other hand, if and are linearly dependent, then , as shown in the first proof.
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 .
Titu's lemma (named after Titu Andreescu, also known as T2 Lemma, Engel's form, or Sedrakyan's inequality) states that for positive reals, we have
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)Edit
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)Edit
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.
A generalization of this is the Hölder inequality.
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 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, as is done when extracting a metric from quantum fidelity.
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
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 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:
The next two theorems are further examples in operator algebra.
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 (Modified Schwarz inequality for 2-positive maps): 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) For reals ,
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