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. It has a number of generalizations, among them Hölder's inequality.
- Statement of the inequality
- First proof
- Second proof
- More proofs
- R2 ordinary two dimensional space
- Rn n dimensional Euclidean space
- Probability theory
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 inequality
The Cauchy–Schwarz inequality states that for all vectors
Moreover, the two sides are equal if and only if
and that equality holds only when either
Then, by linearity of the inner product in its first argument, one has
and, after multiplication by
In the special case that
Now assume that the special case above does not hold: that
There are indeed 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 ℝ and not ℂ.
R2 (ordinary two-dimensional space)
In the usual 2-dimensional space with the dot product, let
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
where equality holds if and only if the vector
Rn (n-dimensional Euclidean space)
In Euclidean space
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
which yields 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 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.
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 or the real part of the right-hand side, as is done when extracting a metric from quantum fidelity.
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,
then the Cauchy–Schwarz inequality becomes
To prove the covariance inequality using the Cauchy–Schwarz inequality, let
where Var denotes variance and Cov denotes covariance.
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
which extends verbatim to positive functionals on C*-algebras:
Theorem (Cauchy–Schwarz inequality for positive functionals on C*-algebras) If
The next two theorems are further examples in operator algebra.
Theorem (Kadison–Schwarz inequality, named after Richard Kadison) If
This extends the fact
Theorem (Modified Schwarz inequality for 2-positive maps). For a 2-positive map