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 or a
complex vector space 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-Man ...
called an inner product. The inner product of two vectors in the space is a
scalar, 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 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 ...
s, and
orthogonality (zero inner product) of vectors. Inner product spaces generalize
Euclidean vector spaces, in which the inner product is the
dot product or ''scalar product'' of
Cartesian coordinates. Inner product spaces of infinite
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
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, norm, topology, etc.) and the linear functions defi ...
. Inner product spaces over the
field 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 fo ...
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, in 1898.
An inner product naturally induces an associated
norm, (denoted
and
in the picture); so, every inner product space is a
normed vector space. 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) 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 of that of
and
is
dense in
for 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 ...
defined by the norm.
Definition
In this article, denotes a
field 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 ...
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 fo ...
s
A
scalar is thus an element of . A bar over an expression representing a scalar denotes the
complex conjugate of this scalar. A zero vector is denoted
for distinguishing it from the scalar .
An ''inner product'' space is a
vector space 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 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.
*
where
denotes the
real part 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''. The
binomial expansion 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 ...
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, '' ...
, 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 ...
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 fo ...
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 is an inner product space, an example of a
Euclidean vector space.
where
is the
transpose of
A function
is an inner product on
if and only if there exists a
symmetric positive-definite matrix 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 positive-definite matrix and
is the
conjugate transpose of
For the real case, this corresponds to the dot product of the results of directionally-different
scaling of the two vectors, with positive
scale factors 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 has several examples of inner product spaces, wherein the metric induced by the inner product yields a
complete metric space. 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 for the norm induced by the preceding inner product, which does not converge to a function.
Random variables
For real
random variables
and
the
expected value of their product