The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality)
is an upper bound on the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
of the
inner product between two
vectors in an inner product space in terms of the product of the vector
norms. It is considered one of the most important and widely used
inequalities in mathematics.
Inner products of vectors can describe
finite sums (via finite-dimensional vector spaces),
infinite series (via vectors in
sequence spaces), and
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
s (via vectors in
Hilbert spaces). 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
and
of an
inner product space
where
is the
inner product. 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 (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
; see the
examples in inner product. Every inner product gives rise to a Euclidean
norm, called the or
, where the norm of a vector
is denoted and defined by
where
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 in terms of the norm:
Moreover, the two sides are equal if and only if
and
are
linearly dependent.
Special cases
Sedrakyan's lemma – positive real numbers
Sedrakyan's inequality, also known as
Bergström's inequality,
Engel's form,
Titu's lemma (or the T2 lemma), states that for real numbers
and positive real numbers
:
or, using summation notation,
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 (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
on
upon substituting
and
. This form is especially helpful when the inequality involves fractions where the numerator is a
perfect square.
- The plane

The real vector space
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, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
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 (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
.
If
and
then the Cauchy–Schwarz inequality becomes:
where
is the
angle
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
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.
: ''n''-dimensional Euclidean space
In
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
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 (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
, the Cauchy–Schwarz inequality becomes:
The Cauchy–Schwarz inequality can be proved using only elementary algebra in this case by observing that the difference of the right and the left hand side is
or by considering the following
quadratic polynomial in
Since the latter polynomial is nonnegative, it has at most one real root, hence its
discriminant is less than or equal to zero. That is,
: ''n''-dimensional complex space
If
with
and
(where
and
) and if the inner product on the vector space
is the canonical complex inner product (defined by
where the bar notation is used for
complex conjugation), then the inequality may be restated more explicitly as follows:
That is,
For the inner product space of
square-integrable complex-valued
functions, the following inequality holds.
The
Hölder inequality is a generalization of this.
Applications
Analysis
In any
inner product space, the
triangle inequality is a consequence of the Cauchy–Schwarz inequality, as is now shown:
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
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
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 (real)
Hilbert spaces are simply generalizations of the
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. 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
and
be
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s. 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
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
and
denotes
covariance.
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
and not
This section gives two proofs of the following theorem:
In both 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
) is the same. It is presented immediately below only once to reduce repetition. It also includes the easy part of the proof of the
Equality Characterization given above; that is, it proves that if
and
are linearly dependent then
By definition,
and
are linearly dependent if and only if one is a scalar multiple of the other.
If
where
is some scalar then
which shows that equality holds in the .
The case where
for some scalar
follows from the previous case:
In particular, if at least one of
and
is the zero vector then
and
are necessarily linearly dependent (for example, if
then
where
), so the above computation shows that the Cauchy–Schwarz inequality holds in this case.
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 via the Pythagorean theorem
The special case of
was proven above so it is henceforth assumed that
Let
It follows from the linearity of the inner product in its first argument that:
Therefore,
is a vector orthogonal to the vector
(Indeed,
is the
projection of
onto the plane orthogonal to
) We can thus apply the
Pythagorean theorem to
which gives
The Cauchy–Schwarz inequality follows by multiplying by
and then taking the square root.
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
The converse was proved at the beginning of this section, so the proof is complete.
Proof by analyzing a quadratic
Consider an arbitrary pair of vectors
. Define the function
defined by
, where
is a complex number satisfying
and
.
Such an
exists since if
then
can be taken to be 1.
Since the inner product is positive-definite,
only takes non-negative real values. On the other hand,
can be expanded using the bilinearity of the inner product:
Thus,
is a polynomial of degree
(unless
which is a case that was checked earlier). Since the sign of
does not change, the discriminant of this polynomial must be non-positive:
The conclusion follows.
For the equality case, notice that
happens if and only if
If
then
and hence
Generalizations
Various generalizations of the Cauchy–Schwarz inequality exist.
Hölder's inequality
In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality (mathematics), inequality between Lebesgue integration, integrals and an indispensable tool for the study of Lp space, spaces.
The numbers an ...
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
In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
(Namely, when the space is a
Hilbert 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:
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, that is,
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
In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality (mathematics), inequality between Lebesgue integration, integrals and an indispensable tool for the study of Lp space, spaces.
The numbers an ...
. There are also non-commutative versions for operators and tensor products of matrices.
Several matrix versions of the Cauchy–Schwarz inequality and
Kantorovich inequality are applied to linear regression models.
See also
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Notes
Citations
References
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External links
Earliest Uses: The entry on the Cauchy–Schwarz inequality has some historical information.Tutorial and Interactive program.
{{DEFAULTSORT:Cauchy-Schwarz inequality
Augustin-Louis Cauchy
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities