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The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality for integrals was published by and . Schwarz gave the modern proof of the integral version.


Statement of the inequality

The Cauchy–Schwarz inequality states that for all vectors \mathbf and \mathbf of an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
it is true that where \langle \cdot, \cdot \rangle is the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. Examples of inner products include the real and complex
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
; see the examples in inner product. Every inner product gives rise to a norm, called the or , where the norm of a vector \mathbf is denoted and defined by: \, \mathbf\, := \sqrt so that this norm and the inner product are related by the defining condition \, \mathbf\, ^2 = \langle \mathbf, \mathbf \rangle, where \langle \mathbf, \mathbf \rangle is always a non-negative real number (even if the inner product is complex-valued). By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality can be written in its more familiar form: Moreover, the two sides are equal if and only if \mathbf and \mathbf are linearly dependent.


Special cases


Sedrakyan's lemma - Positive real numbers

Sedrakyan's inequality, also called Bergström's inequality, Engel's form, the T2 lemma, or
Titu Titu () is a town in Dâmbovița County, Muntenia, Romania, with a population of 9,658 . Location The town in located in the southern part of the county, in the center of the Wallachian Plain. It lies at a distance of from the county seat, Tâ ...
's lemma, states that for real numbers u_1, u_2, \dots, u_n and positive real numbers v_1, v_2, \dots, v_n: \frac \leq \sum_^n \frac \quad \text \quad \frac \leq \frac + \frac + \cdots + \frac . It is a direct consequence of the Cauchy–Schwarz inequality, obtained by using the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
on \R^n upon substituting u_i' = u_i / \sqrt and v_i' = \sqrt. This form is especially helpful when the inequality involves fractions where the numerator is a perfect square.


R2 - The plane

The real vector space \R^2 denotes the 2-dimensional plane. It is also the 2-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
where the inner product is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
. If \mathbf = \left(u_1, u_2\right) and \mathbf = \left(v_1, v_2\right) then the Cauchy–Schwarz inequality becomes: \langle \mathbf, \mathbf \rangle^2 = (\, \mathbf\, \, \mathbf\, \cos \theta)^2 \leq \, \mathbf\, ^2 \, \mathbf\, ^2, where \theta is the
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
between \mathbf and \mathbf. 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 u_1, u_2, v_1, and v_2 as \left(u_1 v_1 + u_2 v_2\right)^2 \leq \left(u_1^2 + u_2^2\right) \left(v_1^2 + v_2^2\right), where equality holds if and only if the vector \left(u_1, u_2\right) is in the same or opposite direction as the vector \left(v_1, v_2\right), or if one of them is the zero vector.


R''n'' - ''n''-dimensional Euclidean space

In
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
\R^n with the standard inner product, which is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
, the Cauchy–Schwarz inequality becomes: \left(\sum_^n u_i v_i\right)^2 \leq \left(\sum_^n u_i^2\right) \left(\sum_^n v_i^2\right). The Cauchy–Schwarz inequality can be proved using only ideas from elementary algebra in this case. Consider the following quadratic polynomial in x 0 \leq \left(u_1 x + v_1\right)^2 + \cdots + \left(u_n x + v_n\right)^2 = \left(\sum_i u_i^2\right) x^2 + 2 \left(\sum_i u_i v_i\right) x + \sum_i v_i^2. Since it is nonnegative, it has at most one real root for x, hence its
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
is less than or equal to zero. That is, \left(\sum_i u_i v_i\right)^2 - \left(\sum_i \right) \left(\sum_i \right) \leq 0,


C''n'' - ''n''-dimensional Complex space

If \mathbf, \mathbf \in \Complex^n with \mathbf = \left(u_1, \ldots, u_n\right) and \mathbf = \left(v_1, \ldots, v_n\right) (where u_1, \ldots, u_n \in \Complex and v_1, \ldots, v_n \in \Complex) and if the inner product on the vector space \Complex^n is the canonical complex inner product (defined by \langle \mathbf, \mathbf \rangle := u_1 \overline + \cdots + u_ \overline, where the bar notation is used for complex conjugation), then the inequality may be restated more explicitly as follows: , \langle \mathbf, \mathbf \rangle, ^2 = \left, \sum_^n u_k\bar_k\^2 \leq \langle \mathbf, \mathbf \rangle \langle \mathbf, \mathbf \rangle = \left(\sum_^n u_k \bar_k\right) \left(\sum_^n v_k \bar_k\right) = \sum_^n \left, u_j\^2 \sum_^n \left, v_k\^2. That is, \left, u_1 \bar_1 + \cdots + u_n \bar_n\^2 \leq \left(\left, u_1\^2 + \cdots + \left, u_n\^2\right) \left(\left, v_1\^2 + \cdots + \left, v_n\^2\right).


''L''2

For the inner product space of square-integrable complex-valued functions, the following inequality: \left, \int_ f(x) \overline\,dx\^2 \leq \int_ , f(x), ^2\,dx \int_ , g(x), ^2 \,dx. The Hölder inequality is a generalization of this.


Applications


Analysis

In any
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, the triangle inequality is a consequence of the Cauchy–Schwarz inequality, as is now shown: \begin \, \mathbf + \mathbf\, ^2 &= \langle \mathbf + \mathbf, \mathbf + \mathbf \rangle && \\ &= \, \mathbf\, ^2 + \langle \mathbf, \mathbf \rangle + \langle \mathbf, \mathbf \rangle + \, \mathbf\, ^2 ~ && ~ \text \langle \mathbf, \mathbf \rangle = \overline \\ &= \, \mathbf\, ^2 + 2 \operatorname \langle \mathbf, \mathbf \rangle + \, \mathbf\, ^2 && \\ &\leq \, \mathbf\, ^2 + 2, \langle \mathbf, \mathbf \rangle, + \, \mathbf\, ^2 && \\ &\leq \, \mathbf\, ^2 + 2\, \mathbf\, \, \mathbf\, + \, \mathbf\, ^2 ~ && ~ \text\\ &\leq(\, \mathbf\, + \, \mathbf\, )^2. && \end Taking square roots gives the triangle inequality: \, \mathbf + \mathbf\, \leq \, \mathbf\, + \, \mathbf\, . The Cauchy–Schwarz inequality is used to prove that the inner product is a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
with respect to the
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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: \cos\theta_ = \frac. 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 (real)
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s are simply generalizations of the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
. It can also be used to define an angle in complex
inner-product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
s, by taking the absolute value or the real part of the right-hand side, as is done when extracting a metric from
quantum fidelity In quantum mechanics, notably in quantum information theory, fidelity is a measure of the "closeness" of two quantum states. It expresses the probability that one state will pass a test to identify as the other. The fidelity is not a metric (mathem ...
.


Probability theory

Let X and Y be
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s, then the covariance inequality is given by: \operatorname(Y) \geq \frac. After defining an inner product on the set of random variables using the expectation of their product, \langle X, Y \rangle := \operatorname(X Y), the Cauchy–Schwarz inequality becomes , \operatorname(XY), ^2 \leq \operatorname(X^2) \operatorname(Y^2). To prove the covariance inequality using the Cauchy–Schwarz inequality, let \mu = \operatorname(X) and \nu = \operatorname(Y), then \begin , \operatorname(X, Y), ^2 &= , \operatorname((X - \mu)(Y - \nu)), ^2 \\ &= , \langle X - \mu, Y - \nu \rangle , ^2\\ &\leq \langle X - \mu, X - \mu \rangle \langle Y - \nu, Y - \nu \rangle \\ & = \operatorname\left((X - \mu)^2\right) \operatorname\left((Y - \nu)^2\right) \\ & = \operatorname(X) \operatorname(Y), \end where \operatorname denotes
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
and \operatorname denotes
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
.


Proofs

There are many different proofs of the Cauchy–Schwarz inequality other than those given below. 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 \mathbb R and not \mathbb C. This section gives proofs of the following theorem: In all of the proofs given below, the proof in the trivial case where at least one of the vectors is zero (or equivalently, in the case where \, \mathbf\, \, \mathbf\, = 0) is the same. It is presented immediately below only once to reduce repetition. It also includes the easy part of the proof the Equality Characterization given above; that is, it proves that if \mathbf and \mathbf are linearly dependent then \left, \langle \mathbf, \mathbf \rangle\ = \, \mathbf\, \, \mathbf\, . By definition, \mathbf and \mathbf are linearly dependent if and only if one is a scalar multiple of the other. If \mathbf = c \mathbf where c is some scalar then , \langle \mathbf, \mathbf \rangle, = , \langle c \mathbf, \mathbf \rangle, = , c \langle \mathbf, \mathbf \rangle, = , c, \, \mathbf\, \, \mathbf\, =\, c \mathbf\, \, \mathbf\, =\, \mathbf\, \, \mathbf\, which shows that equality holds in the . The case where \mathbf = c \mathbf for some scalar c is very similar, with the main difference between the complex conjugation of c: , \langle \mathbf, \mathbf \rangle, = , \langle \mathbf, c \mathbf \rangle, = \left, \overline \langle \mathbf, \mathbf \rangle\ = \left, \overline\ \, \mathbf\, \, \mathbf\, = , c, \, \mathbf\, \, \mathbf\, =\, \mathbf\, \, c \mathbf\, =\, \mathbf\, \, \mathbf\, . If at least one of \mathbf and \mathbf is the zero vector then \mathbf and \mathbf are necessarily linearly dependent (just scalar multiply the non-zero vector by the number 0 to get the zero vector; for example, if \mathbf = \mathbf then let c = 0 so that \mathbf = c \mathbf), which proves the converse of this characterization in this special case; that is, this shows that if at least one of \mathbf and \mathbf is \mathbf then the Equality Characterization holds. If \mathbf = \mathbf, which happens if and only if \, \mathbf\, = 0, then \, \mathbf\, \, \mathbf\, = 0 and , \langle \mathbf, \mathbf \rangle, = , \langle \mathbf, \mathbf \rangle, = , 0, = 0 so that in particular, the Cauchy-Schwarz inequality holds because both sides of it are 0. The proof in the case of \mathbf = \mathbf is identical. Consequently, the Cauchy-Schwarz inequality only needs to be proven only for non-zero vectors and also only the non-trivial direction of the Equality Characterization must be shown.


Proof 1

The special case of \mathbf = \mathbf was proven above so it is henceforth assumed that \mathbf \neq \mathbf. The Cauchy–Schwarz equality (and the rest of the theorem) is an almost immediate corollary of the following : Equality is readily verified by elementarily expanding \left\, \, \mathbf\, ^2 \mathbf - \langle \mathbf, \mathbf \rangle \mathbf\right\, ^2 (via the definition of the norm) and then simplifying: This expansion does not require \mathbf to be non-zero; however, \mathbf must be non-zero in order to divide both sides by \, \mathbf\, ^2 and to deduce the Cauchy-Schwarz inequality from it. Swapping \mathbf and \mathbf gives rise to: \left\, \, \mathbf\, ^2 \mathbf - \overline \mathbf\right\, ^2 ~=~ \, \mathbf\, ^2 \left \mathbf\, ^2\, \mathbf\, ^2 - , \langle \mathbf, \mathbf \rangle, ^2\right/math> and thus \begin \, \mathbf\, ^2\, \mathbf\, ^2 \left \mathbf\, ^2 \, \mathbf\, ^2 - , \langle \mathbf, \mathbf \rangle, ^2\right~&=~ \, \mathbf\, ^2 \left\, \, \mathbf\, ^2 \mathbf - \langle \mathbf, \mathbf \rangle \mathbf\right\, ^2 \\ ~&=~ \, \mathbf\, ^2 \left\, \, \mathbf\, ^2 \mathbf - \overline \mathbf\right\, ^2. \\ \end


Proof 2

The special case of \mathbf = \mathbf was proven above so it is henceforth assumed that \mathbf \neq \mathbf. Let \mathbf := \mathbf - \frac \mathbf. It follows from the linearity of the inner product in its first argument that: \langle \mathbf, \mathbf \rangle = \left\langle \mathbf - \frac \mathbf, \mathbf \right\rangle = \langle \mathbf, \mathbf \rangle - \frac \langle \mathbf, \mathbf \rangle = 0. Therefore, \mathbf is a vector orthogonal to the vector \mathbf (Indeed, \mathbf is the projection of \mathbf onto the plane orthogonal to \mathbf.) We can thus apply the Pythagorean theorem to \mathbf= \frac \mathbf + \mathbf which gives \, \mathbf\, ^2 = \left, \frac\^2 \, \mathbf\, ^2 + \, \mathbf\, ^2 = \frac \,\, \mathbf\, ^2 + \, \mathbf\, ^2 = \frac + \, \mathbf\, ^2 \geq \frac. The Cauchy–Schwarz inequality follows by multiplying by \, \mathbf\, ^2 and then taking the square root. Moreover, if the relation \geq in the above expression is actually an equality, then \, \mathbf\, ^2 = 0 and hence \mathbf = \mathbf; the definition of \mathbf then establishes a relation of linear dependence between \mathbf and \mathbf. The converse was proved at the beginning of this section, so the proof is complete. \blacksquare


Proof for real inner products

Let (V, \langle \cdot, \cdot \rangle) be a real inner product space. Consider an arbitrary pair \mathbf, \mathbf \in V and the function p : \R \to \R defined by p(t) = \langle t\mathbf + \mathbf, t\mathbf + \mathbf\rangle. Since the inner product is positive-definite, p(t) only takes non-negative values. On the other hand, p(t) can be expanded using the bilinearity of the inner product and using the fact that \langle \mathbf, \mathbf \rangle = \langle \mathbf, \mathbf \rangle for real inner products: p(t) = \Vert \mathbf \Vert^2 t^2 + t \left langle \mathbf, \mathbf \rangle + \langle \mathbf, \mathbf \rangle\right+ \Vert \mathbf \Vert^2 = \Vert \mathbf \Vert^2 t^2 + 2t \langle \mathbf, \mathbf \rangle + \Vert \mathbf \Vert^2. Thus, p is a polynomial of degree 2 (unless u = 0, which is a case that can be independently verified). Since the sign of p does not change, the discriminant of this polynomial must be non-positive: \Delta = 4 \left(\langle \mathbf, \mathbf \rangle ^2 - \Vert \mathbf \Vert^2 \Vert \mathbf \Vert^2\right) \leqslant 0. The conclusion follows. For the equality case, notice that \Delta = 0 happens if and only if p(t) = (t\Vert \mathbf \Vert + \Vert \mathbf \Vert)^2. If t_0 = -\Vert \mathbf \Vert / \Vert \mathbf \Vert, then p(t_0) = \langle t_0\mathbf + \mathbf,t_0\mathbf + \mathbf\rangle = 0, and hence \mathbf = -t_0\mathbf.


Proof for the dot product

The Cauchy-Schwarz inequality in the case where the inner product is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
on \R^n is now proven. The Cauchy-Schwarz inequality may be rewritten as \left, \mathbf \cdot \mathbf\^2 \leq \left\, \mathbf\right\, ^2 \, \left\, \mathbf\right\, ^2 or equivalently, \left(\mathbf \cdot \mathbf\right)^2 \leq \left(\mathbf \cdot \mathbf\right) \, \left(\mathbf \cdot \mathbf\right) for \mathbf := \left(a_1, \ldots, a_n\right), \mathbf := \left(b_1, \ldots, b_n\right) \in \R^n, which expands to: \left(a_1^2 + a_2^2 + \cdots + a_n^2\right) \left(b_1^2 + b_2^2 + \cdots + b_n^2\right) \geq \left(a_1b_1 + a_2b_2 + \cdots + a_nb_n\right)^2. To simplify, let \begin A &= a_1^2 + a_2^2 + \cdots + a_n^2, \\ B &= b_1^2 + b_2^2 + \cdots + b_n^2 \\ D &= a_1 b_1 + a_2 b_2 + \cdots + a_n b_n \\\end so that the statement that remains to be to proven can be written as A B \geq D^2, which can be rearranged to D^2 - A B \leq 0. The
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
of the
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
A x^2 + 2 D x + B is 4 D^2 - 4 A B. Therefore, to complete the proof it is sufficient to prove that this quadratic either has no real roots or exactly one real root, because this will imply: 4 \left(D^2 - A B\right) \leq 0. Substituting the values of A, B, D into A x^2 + 2 D x + B gives: \begin A x^2 + 2 D x + B &= \left(a_1^2 + a_2^2 + \cdots + a_n^2\right) x^2 + 2 \left(a_1 b_1 + a_2 b_2 + \cdots + a_n b_n\right) x + \left(b_1^2 + b_2^2 + \cdots + b_n^2\right) \\ &= \left(a_1^2 x^2 + 2a_1 b_1 x + b_1^2\right) + \left(a_2^2 x^2 + 2a_2 b_2 x + b_2^2\right) + \cdots + \left(a_n^2 x^2 + 2a_n b_n x + b_n^2\right) \\ &= \left(a_1x + b_1\right)^2 + \left(a_2 x + b_2\right)^2 + \cdots + \left(a_n x + b_n\right)^2 \\ &\geq 0 \end which is a sum of terms that are each \,\geq 0\, by th
trivial inequality:
r^2 \geq 0 for all r \in \R. This proves the inequality and so to finish the proof, it remains to show that equality is achievable. The equality a_i x = - b_i is the equality case for Cauchy-Schwarz after inspecting \left(a_1 x + b_1\right)^2 + \left(a_2 x + b_2\right)^2 + \cdots + \left(a_n x + b_n\right)^2 \geq 0, which proves that equality is achievable. \blacksquare


Generalizations

Various generalizations of the Cauchy–Schwarz inequality exist. Hölder's inequality generalizes it to L^p 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 In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
(Namely, when the space is a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
). 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 L^2(m), m being a finite measure, the standard inner product gives rise to a positive functional \varphi by \varphi (g) = \langle g, 1 \rangle. Conversely, every positive linear functional \varphi on L^2(m) can be used to define an inner product \langle f, g \rangle _\varphi := \varphi\left(g^* f\right), where g^* is the
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
complex conjugate of g. In this language, the Cauchy–Schwarz inequality becomes \left, \varphi\left(g^* f\right)\^2 \leq \varphi\left(f^* f\right) \varphi\left(g^* g\right), which extends verbatim to positive functionals on C*-algebras: The next two theorems are further examples in operator algebra. This extends the fact \varphi\left(a^*a\right) \cdot 1 \geq \varphi(a)^* \varphi(a) = , \varphi(a), ^2, when \varphi is a linear functional. The case when a is self-adjoint, that is, a = a^*, is sometimes known as Kadison's inequality. Another generalization is a refinement obtained by interpolating between both sides of the Cauchy-Schwarz inequality: This theorem can be deduced from Hölder's inequality. There are also non commutative versions for operators and tensor products of matrices. A survey of matrix versions of Cauchy-Schwarz and Kantorovich inequalities is available.


See also

* * * * * *


Notes


Citations


References

* * * * * * * . * * . * * *


External links


Earliest Uses: The entry on the Cauchy–Schwarz inequality has some historical information.


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