![Product Spaces Drawing (1)](https://upload.wikimedia.org/wikipedia/commons/d/d1/Product_Spaces_Drawing_%281%29.png)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an inner product space (or, rarely, a
Hausdorff pre-Hilbert space) is a
real vector space
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or a
complex vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
with an
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Ma ...
called an inner product. The inner product of two vectors in the space is a
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
, often denoted with
angle brackets
A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
such as in
. Inner products allow formal definitions of intuitive geometric notions, such as lengths,
angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
s, and
orthogonality (zero inner product) of vectors. Inner product spaces generalize
Euclidean vector 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 Euclidean s ...
s, in which 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 algebra ...
or ''scalar product'' of
Cartesian coordinates. Inner product spaces of infinite
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
are widely used in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
. Inner product spaces over the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to
Giuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...
, in 1898.
An inner product naturally induces an associated
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
, (denoted
and
in the picture); so, every inner product space is a
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
. If this normed space is also
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
(that is, 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 ...
) then the inner product space is a
Hilbert space. If an inner product space is not a Hilbert space, it can be ''extended'' by
completion to a Hilbert space
This means that
is a
linear subspace of
the inner product of
is the
restriction
Restriction, restrict or restrictor may refer to:
Science and technology
* restrict, a keyword in the C programming language used in pointer declarations
* Restriction enzyme, a type of enzyme that cleaves genetic material
Mathematics and logi ...
of that of
and
is
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in
for the
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
defined by the norm.
Definition
In this article, denotes a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
that is either the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s
or the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s
A
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
is thus an element of . A bar over an expression representing a scalar denotes the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of this scalar. A zero vector is denoted
for distinguishing it from the scalar .
An ''inner product'' space is a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over the field together with an ''inner product'', that is a map
:
that satisfies the following three properties for all vectors
and all scalars
* ''Conjugate symmetry'':
As
if and only if is real, conjugate symmetry implies that
is always a real number. If is
, conjugate symmetry is just symmetry.
*
Linearity
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
in the first argument:
[By combining the ''linear in the first argument'' property with the ''conjugate symmetry'' property you get ''conjugate-linear in the second argument'': . This is how the inner product was originally defined and is used in most mathematical contexts. A different convention has been adopted in theoretical physics and quantum mechanics, originating in the bra-ket notation of ]Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
, where the inner product is taken to be ''linear in the second argument'' and ''conjugate-linear in the first argument''; this convention is used in many other domains such as engineering and computer science.
*
Positive-definiteness: if is not zero, then
(conjugate symmetry implies that
is real).
If the positive-definiteness condition is replaced by merely requiring that
for all , then one obtains the definition of ''positive semi-definite Hermitian form''. A positive semi-definite Hermitian form
is an inner product if and only if for all ''x'', if
then ''x = 0''.
Basic properties
In the following properties, which result almost immediately from the definition of an inner product, and are arbitrary vectors, and and are arbitrary scalars.
*
*
is real and nonnegative.
*
if and only if
*
This implies that an inner product is a
sesquilinear form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows o ...
.
*
where
denotes the
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
of its argument.
Over
, conjugate-symmetry reduces to symmetry, and sesquilinearity reduces to bilinearity. Hence an inner product on a real vector space is a ''positive-definite symmetric
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
''. The
binomial expansion
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
of a square becomes
:
Convention variant
Some authors, especially in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
matrix algebra
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
, prefer to define inner products and sesquilinear forms with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second.
Notation
Several notations are used for inner products, including
,
,
and
, as well as the usual dot product.
Some examples
Real and complex numbers
Among the simplest examples of inner product spaces are
and
The
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s
are a vector space over
that becomes an inner product space with arithmetic multiplication as its inner product:
The
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s
are a vector space over
that becomes an inner product space with the inner product
Unlike with the real numbers, the assignment
does define a complex inner product on
Euclidean vector space
More generally, the
real -space with 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 algebra ...
is an inner product space, an example of a
Euclidean vector 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 Euclidean s ...
.
where
is the
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
of
A function
is an inner product on
if and only if there exists a
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
positive-definite matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a ...
such that
for all
If
is the
identity matrix then
is the dot product. For another example, if
and
is positive-definite (which happens if and only if
and one/both diagonal elements are positive) then for any
As mentioned earlier, every inner product on
is of this form (where
and
satisfy
).
Complex coordinate space
The general form of an inner product on
is known as the
Hermitian form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...
and is given by
where
is any
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
positive-definite matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a ...
and
is the
conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
of
For the real case, this corresponds to the dot product of the results of directionally-different
scaling
Scaling may refer to:
Science and technology
Mathematics and physics
* Scaling (geometry), a linear transformation that enlarges or diminishes objects
* Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
of the two vectors, with positive
scale factor
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions. The result of uniform scaling is similar ...
s and orthogonal directions of scaling. It is a
weighted-sum version of the dot product with positive weights—up to an orthogonal transformation.
Hilbert space
The article on
Hilbert spaces
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 naturally ...
has several examples of inner product spaces, wherein the metric induced by the inner product yields a
complete metric space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
. An example of an inner product space which induces an incomplete metric is the space
of continuous complex valued functions
and
on the interval
The inner product is
This space is not complete; consider for example, for the interval the sequence of continuous "step" functions,
defined by:
This sequence is a
Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
for the norm induced by the preceding inner product, which does not converge to a function.
Random variables
For real
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
and
the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of their product