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
Also known as Cauchy–Bunyakovsky–Schwarz inequality, Cauchy–Schwarz–Bunyakovsky inequality
The Cauchy–Schwarz inequality is an upper bound on the absolute value 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.
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The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value 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 integrals (via vectors in Hilbert spaces). The inequality for sums was published by Augustin-Louis Cauchy (1821). The corresponding inequality for integrals was published by Viktor Bunyakovsky (1859) and Hermann Schwarz (1888). Schwarz gave the modern proof of the integral version.
Contents
Statement of the inequality
The Cauchy–Schwarz inequality states that for all vectors
u
{\displaystyle \mathbf {u} }
and
v
{\displaystyle \mathbf {v} }
of an inner product space
where
⟨
⋅
,
⋅
⟩
{\displaystyle \langle \cdot ,\cdot \rangle }
is the inner product. Examples of inner products include the real and complex dot product; see the examples in inner product. Every inner product gives rise to a Euclidean
ℓ
2
{\displaystyle \ell _{2}}
norm, called the canonical or induced norm, where the norm of a vector
u
{\displaystyle \mathbf {u} }
is denoted and defined by
‖
u
‖
:=
⟨
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
u
1
,
u
2
,
…
,
u
n
{\displaystyle u_{1},u_{2},\dots ,u_{n}}
and positive real numbers
v
1
,
v
2
,
…
,
v
n
{\displaystyle v_{1},v_{2},\dots ,v_{n}}
:
(
u
1
+
u
2
+
⋯
R2 - The plane
The real vector space
R
2
{\displaystyle \mathbb {R} ^{2}}
denotes the 2-dimensional plane. It is also the 2-dimensional Euclidean space where the inner product is the dot product.
If
u
=
(
u
1
,
u
2
)
{\displaystyle \mathbf {u} =(u_{1},u_{2})}
and
v
=
(
v
1
,
v
2
)
{\displaystyle \mathbf {v} =(v_{1},v_{2})}
then the Cauchy–Schwarz inequality becomes:
⟨
u
,
v
Rn: n-dimensional Euclidean space
In Euclidean space
R
n
{\displaystyle \mathbb {R} ^{n}}
with the standard inner product, which is the dot product, the Cauchy–Schwarz inequality becomes:
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.
Linear algebra
The Cauchy-Schwarz inequality can be used to prove the spectral theorem for self-adjoint operators in the finite-dimensional case.
Let
A
{\displaystyle A}
be a self-adjoint operator on a finite-dimensional inner product space and
After defining an inner product on the set of random variables using the expectation of their product,
⟨
X
,
Graph theory
In extremal graph theory, the Cauchy-Schwarz inequality is used to prove Mantel's theorem, which states that the number of edges in a triangle-free graph on
respectively. Since the graph is triangle-free, the neighbourhoods of
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
R
{\displaystyle \mathbb {R} }
and not
C
.
{\displaystyle \mathbb {C} .}
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
Various generalizations of the Cauchy–Schwarz inequality exist. Hölder's inequality generalizes it to
L
p
{\displaystyle 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 (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
L
2
(
m
)
,
m
{\displaystyle L^{2}(m),m}
being a finite measure, the standard inner product gives rise to a positive functional
φ
{\displaystyle \varphi }
by
φ
(
g
)
=
⟨
g
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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:
The form presented here is perhaps the easiest in which to understand the inequality, as 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
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
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.
hence equality holds. By the characterization of equality,
u
{\displaystyle \mathbf {u} }
and
A
2
u
{\displaystyle A^{2}\mathbf {u} }
are linearly dependent, therefore
u
{\displaystyle \mathbf {u} }
is an eigenvector of
A
2
{\displaystyle A^{2}}
. From here it is straightforward to deduce that
A
{\displaystyle A}
has an eigenvector, then the spectral theorem follows by taking the orthogonal complement and arguing by induction on the dimension of the inner product space.
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.
