Inner Product
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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 \langle a, b \rangle. 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 In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
(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 A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
. 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 stand ...
, 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 , x, and , y, 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" i ...
. 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 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 ...
. If an inner product space is not a Hilbert space, it can be ''extended'' by completion to a Hilbert space \overline. This means that H is a
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
of \overline, the inner product of H 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 \overline, and H 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 \overline 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 \R, 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 \Complex. 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 \mathbf 0 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 : \langle \cdot, \cdot \rangle : V \times V \to F that satisfies the following three properties for all vectors x,y,z\in V and all scalars * ''Conjugate symmetry'': \langle x, y \rangle = \overline. As a = \overline if and only if is real, conjugate symmetry implies that \langle x, x \rangle is always a real number. If is \R, 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 r ...
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'': \langle x,by \rangle = \langle x,y \rangle \overline . 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.
\langle ax+by, z \rangle = a \langle x, z \rangle + b \langle y, z \rangle. * Positive-definiteness: if is not zero, then \langle x, x \rangle > 0 (conjugate symmetry implies that \langle x, x \rangle is real). If the positive-definiteness condition is replaced by merely requiring that \langle x, x \rangle \geq 0 for all , then one obtains the definition of ''positive semi-definite Hermitian form''. A positive semi-definite Hermitian form \langle \cdot, \cdot \rangle is an inner product if and only if for all ''x'', if \langle x, x \rangle = 0 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. *\langle \mathbf, x \rangle=\langle x,\mathbf\rangle=0. * \langle x, x \rangle is real and nonnegative. *\langle x, x \rangle = 0 if and only if x=\mathbf. *\langle x, ay+bz \rangle= \overline a \langle x, y \rangle + \overline b \langle x, z \rangle.
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 ...
. *\langle x + y, x + y \rangle = \langle x, x \rangle + 2\operatorname(\langle x, y \rangle) + \langle y, y \rangle, where \operatorname
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 \R, 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 : \langle x + y, x + y \rangle = \langle x, x \rangle + 2\langle x, y \rangle + \langle y, y \rangle .


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 \langle \cdot, \cdot \rangle , \left ( \cdot, \cdot \right ) , \langle \cdot , \cdot \rangle and \left ( \cdot , \cdot \right ) , as well as the usual dot product.


Some examples


Real and complex numbers

Among the simplest examples of inner product spaces are \R and \Complex. 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 \R are a vector space over \R that becomes an inner product space with arithmetic multiplication as its inner product: \langle x, y \rangle := x y \quad \text x, y \in \R. 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 \Complex are a vector space over \Complex that becomes an inner product space with the inner product \langle x, y \rangle := x \overline \quad \text x, y \in \Complex. Unlike with the real numbers, the assignment (x, y) \mapsto x y does define a complex inner product on \Complex.


Euclidean vector space

More generally, the real n-space \R^n 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 ...
. \left\langle \begin x_1 \\ \vdots \\ x_n \end, \begin y_1 \\ \vdots \\ y_n \end \right\rangle = x^\textsf y = \sum_^n x_i y_i = x_1 y_1 + \cdots + x_n y_n, where x^ 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 x. A function \langle \,\cdot, \cdot\, \rangle : \R^n \times \R^n \to \R is an inner product on \R^n 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 co ...
\mathbf such that \langle x, y \rangle = x^ \mathbf y for all x, y \in \R^n. If \mathbf is the
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
then \langle x, y \rangle = x^ \mathbf y is the dot product. For another example, if n = 2 and \mathbf = \begin a & b \\ b & d \end is positive-definite (which happens if and only if \det \mathbf = a d - b^2 > 0 and one/both diagonal elements are positive) then for any x := \left _1, x_2\right, y := \left _1, y_2\right \in \R^2, \langle x, y \rangle := x^ \mathbf y = \left _1, x_2\right\begin a & b \\ b & d \end \begin y_1 \\ y_2 \end = a x_1 y_1 + b x_1 y_2 + b x_2 y_1 + d x_2 y_2. As mentioned earlier, every inner product on \R^2 is of this form (where b \in \R, a > 0 and d > 0 satisfy a d > b^2).


Complex coordinate space

The general form of an inner product on \Complex^n 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 \langle x, y \rangle = y^\dagger \mathbf x = \overline, where M 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 co ...
and y^ 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 con ...
of y. 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 boun ...
. An example of an inner product space which induces an incomplete metric is the space C( , b of continuous complex valued functions f and g on the interval , b The inner product is \langle f, g \rangle = \int_a^b f(t) \overline \, \mathrmt. This space is not complete; consider for example, for the interval the sequence of continuous "step" functions, \_k, defined by: f_k(t) = \begin 0 & t \in 1, 0\\ 1 & t \in \left tfrac, 1\right\\ kt & t \in \left(0, \tfrac\right) \end This sequence is a
Cauchy sequence 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 m ...
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 X and Y, 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 \langle X, Y \rangle = \mathbb Y/math> is an inner product. In this case, \langle X, X \rangle = 0 if and only if \mathbb = 0= 1 (that is, X = 0
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
), where \mathbb denotes the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
of the event. This definition of expectation as inner product can be extended to
random vector In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value. ...
s as well.


Complex matrices

The inner product for complex square matrices of the same size is the
Frobenius inner product In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar. It is often denoted \langle \mathbf,\mathbf \rangle_\mathrm. The operation is a component-wise inner product of two matrices as though t ...
\langle A, B \rangle := \operatorname\left(AB^\right). Since trace and transposition are linear and the conjugation is on the second matrix, it is a sesquilinear operator. We further get Hermitian symmetry by, \langle A, B \rangle = \operatorname\left(AB^\right) = \overline = \overline Finally, since for A nonzero, \langle A, A\rangle = \sum_ \left, A_\^2 > 0 , we get that the Frobenius inner product is positive definite too, and so is an inner product.


Vector spaces with forms

On an inner product space, or more generally a vector space with a
nondegenerate form In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent definiti ...
(hence an isomorphism V \to V^*), vectors can be sent to covectors (in coordinates, via transpose), so that one can take the inner product and outer product of two vectors—not simply of a vector and a covector.


Basic results, terminology, and definitions


Norm properties

Every inner product space induces a
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 ...
, called its , that is defined by \, x\, = \sqrt. With this norm, every inner product space becomes 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" i ...
. So, every general property of normed vector spaces applies to inner product spaces. In particular, one has the following properties:


Orthogonality


Real and complex parts of inner products

Suppose that \langle \cdot, \cdot \rangle is an inner product on V (so it is antilinear in its second argument). The
polarization identity In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then t ...
shows that 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 the inner product is \operatorname \langle x, y \rangle = \frac \left(\, x + y\, ^2 - \, x - y\, ^2\right). If V is a real vector space then \langle x, y \rangle = \operatorname \langle x, y \rangle = \frac \left(\, x + y\, ^2 - \, x - y\, ^2\right) and the
imaginary 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 ...
(also called the ) of \langle \cdot, \cdot \rangle is always 0. Assume for the rest of this section that V is a complex vector space. The
polarization identity In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then t ...
for complex vector spaces shows that :\begin \langle x, \ y \rangle &= \frac \left(\, x + y\, ^2 - \, x - y\, ^2 + i\, x + iy\, ^2 - i\, x - iy\, ^2 \right) \\ &= \operatorname \langle x, y \rangle + i \operatorname \langle x, i y \rangle. \\ \end The map defined by \langle x \mid y \rangle = \langle y, x \rangle for all x, y \in V satisfies the axioms of the inner product except that it is antilinear in its , rather than its second, argument. The real part of both \langle x \mid y \rangle and \langle x, y \rangle are equal to \operatorname \langle x, y \rangle but the inner products differ in their complex part: :\begin \langle x \mid y \rangle &= \frac \left(\, x + y\, ^2 - \, x - y\, ^2 - i\, x + iy\, ^2 + i\, x - iy\, ^2 \right) \\ &= \operatorname \langle x, y \rangle - i \operatorname \langle x, i y \rangle. \\ \end The last equality is similar to the formula expressing a linear functional in terms of its real part. These formulas show that every complex inner product is completely determined by its real part. Moreover, this real part defines an inner product on V, considered as a real vector space. There is thus a one-to-one correspondence between complex inner products on a complex vector space V, and real inner products on V. For example, suppose that V = \Complex^n for some integer n > 0. When V is considered as a real vector space in the usual way (meaning that it is identified with the 2 n-dimensional real vector space \R^, with each \left(a_1 + i b_1, \ldots, a_n + i b_n\right) \in \Complex^n identified with \left(a_1, b_1, \ldots, a_n, b_n\right) \in \R^), then 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 ...
x \,\cdot\, y = \left(x_1, \ldots, x_\right) \, \cdot \, \left(y_1, \ldots, y_\right) := x_1 y_1 + \cdots + x_ y_ defines a real inner product on this space. The unique complex inner product \langle \,\cdot, \cdot\, \rangle on V = \C^n induced by the dot product is the map that sends c = \left(c_1, \ldots, c_n\right), d = \left(d_1, \ldots, d_n\right) \in \Complex^n to \langle c, d \rangle := c_1 \overline + \cdots + c_n \overline (because the real part of this map \langle \,\cdot, \cdot\, \rangle is equal to the dot product). Real vs. complex inner products Let V_ denote V considered as a vector space over the real numbers rather than complex numbers. 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 the complex inner product \langle x, y \rangle is the map \langle x, y \rangle_ = \operatorname \langle x, y \rangle ~:~ V_ \times V_ \to \R, which necessarily forms a real inner product on the real vector space V_. Every inner product on a real vector space is a bilinear and symmetric map. For example, if V = \Complex with inner product \langle x, y \rangle = x \overline, where V is a vector space over the field \Complex, then V_ = \R^2 is a vector space over \R and \langle x, y \rangle_ 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 ...
x \cdot y, where x = a + i b \in V = \Complex is identified with the point (a, b) \in V_ = \R^2 (and similarly for y); thus the standard inner product \langle x, y \rangle = x \overline, on \Complex is an "extension" the dot product . Also, had \langle x, y \rangle been instead defined to be the \langle x, y \rangle = x y (rather than the usual \langle x, y \rangle = x \overline) then its real part \langle x, y \rangle_ would be the dot product; furthermore, without the complex conjugate, if x \in \C but x \not\in \R then \langle x, x \rangle = x x = x^2 \not\in [0, \infty) so the assignment x \mapsto \sqrt would not define a norm. The next examples show that although real and complex inner products have many properties and results in common, they are not entirely interchangeable. For instance, if \langle x, y \rangle = 0 then \langle x, y \rangle_ = 0, but the next example shows that the converse is in general true. Given any x \in V, the vector i x (which is the vector x rotated by 90°) belongs to V and so also belongs to V_ (although scalar multiplication of x by i = \sqrt is not defined in V_, the vector in V denoted by i x is nevertheless still also an element of V_). For the complex inner product, \langle x, ix \rangle = -i \, x\, ^2, whereas for the real inner product the value is always \langle x, ix \rangle_ = 0. If \langle \,\cdot, \cdot\, \rangle is a complex inner product and A : V \to V is a continuous linear operator that satisfies \langle x, A x \rangle = 0 for all x \in V, then A = 0. This statement is no longer true if \langle \,\cdot, \cdot\, \rangle is instead a real inner product, as this next example shows. Suppose that V = \Complex has the inner product \langle x, y \rangle := x \overline mentioned above. Then the map A : V \to V defined by A x = ix is a linear map (linear for both V and V_) that denotes rotation by 90^ in the plane. Because x and A x perpendicular vectors and \langle x, Ax \rangle_ is just the dot product, \langle x, Ax \rangle_ = 0 for all vectors x; nevertheless, this rotation map A is certainly not identically 0. In contrast, using the complex inner product gives \langle x, Ax \rangle = -i \, x\, ^2, which (as expected) is not identically zero.


Orthonormal sequences

Let V be a finite dimensional inner product space of dimension n. Recall that every Basis (linear algebra), basis of V consists of exactly n linearly independent vectors. Using the Gram–Schmidt process we may start with an arbitrary basis and transform it into an orthonormal basis. That is, into a basis in which all the elements are orthogonal and have unit norm. In symbols, a basis \ is orthonormal if \langle e_i, e_j \rangle = 0 for every i \neq j and \langle e_i, e_i \rangle = \, e_a\, ^2 = 1 for each index i. This definition of orthonormal basis generalizes to the case of infinite-dimensional inner product spaces in the following way. Let V be any inner product space. Then a collection E = \left\_ is a for V if the subspace of V generated by finite linear combinations of elements of E is dense in V (in the norm induced by the inner product). Say that E is an for V if it is a basis and \left\langle e_, e_ \right\rangle = 0 if a \neq b and \langle e_a, e_a \rangle = \, e_a\, ^2 = 1 for all a, b \in A. Using an infinite-dimensional analog of the Gram-Schmidt process one may show: Theorem. Any separable inner product space has an orthonormal basis. Using the
Hausdorff maximal principle In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 (Moore 1982:168). It states that in any partially ordered set, every totally ordered subset is contained ...
and the fact that in a complete inner product space orthogonal projection onto linear subspaces is well-defined, one may also show that Theorem. Any complete inner product space has an orthonormal basis. The two previous theorems raise the question of whether all inner product spaces have an orthonormal basis. The answer, it turns out is negative. This is a non-trivial result, and is proved below. The following proof is taken from Halmos's ''A Hilbert Space Problem Book'' (see the references). :
Parseval's identity In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which ...
leads immediately to the following theorem: Theorem. Let V be a separable inner product space and \left\_k an orthonormal basis of V. Then the map x \mapsto \bigl\_ is an isometric linear map V \mapsto \ell^2 with a dense image. This theorem can be regarded as an abstract form of
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
, in which an arbitrary orthonormal basis plays the role of the sequence of
trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...
s. Note that the underlying index set can be taken to be any countable set (and in fact any set whatsoever, provided \ell^2 is defined appropriately, as is explained in the article
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 ...
). In particular, we obtain the following result in the theory of Fourier series: Theorem. Let V be the inner product space C \pi, \pi Then the sequence (indexed on set of all integers) of continuous functions e_k(t) = \frac is an orthonormal basis of the space C \pi, \pi/math> with the L^2 inner product. The mapping f \mapsto \frac \left\_ is an isometric linear map with dense image. Orthogonality of the sequence \_k follows immediately from the fact that if k \neq j, then \int_^\pi e^ \, \mathrmt = 0. Normality of the sequence is by design, that is, the coefficients are so chosen so that the norm comes out to 1. Finally the fact that the sequence has a dense algebraic span, in the , follows from the fact that the sequence has a dense algebraic span, this time in the space of continuous periodic functions on \pi, \pi/math> with the uniform norm. This is the content of the Weierstrass theorem on the uniform density of trigonometric polynomials.


Operators on inner product spaces

Several types of
linear 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 r ...
maps A : V \to W between inner product spaces V and W are of relevance: * : A : V \to W is linear and continuous with respect to the metric defined above, or equivalently, A is linear and the set of non-negative reals \, where x ranges over the closed unit ball of V, is bounded. * : A : V \to W is linear and \langle Ax, y \rangle = \langle x, Ay \rangle for all x, y \in V. * : A : V \to W satisfies \, A x\, = \, x\, for all x \in V. A (resp. an ) is an isometry that is also a linear map (resp. an
antilinear map In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if \begin f(x + y) &= f(x) + f(y) && \qquad \text \\ f(s x) &= \overline f(x) && \qquad \text \\ \end hold for all vectors x, y ...
). For inner product spaces, the
polarization identity In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then t ...
can be used to show that A is an isometry if and only if \langle Ax, Ay \rangle = \langle x, y \rangle for all x, y \in V. All isometries are
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
. The
Mazur–Ulam theorem In mathematics, the Mazur–Ulam theorem states that if V and W are normed spaces over R and the mapping :f\colon V\to W is a surjective isometry, then f is affine. It was proved by Stanisław Mazur and Stanisław Ulam in response to a questi ...
establishes that every surjective isometry between two normed spaces is an
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...
. Consequently, an isometry A between real inner product spaces is a linear map if and only if A(0) = 0. Isometries are
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s between inner product spaces, and morphisms of real inner product spaces are orthogonal transformations (compare with
orthogonal matrix In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity ma ...
). * : A : V \to W is an isometry which is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
(and hence
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
). Isometrical isomorphisms are also known as unitary operators (compare with
unitary matrix In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is ...
). From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. The
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
provides a canonical form for symmetric, unitary and more generally
normal operator In mathematics, especially functional analysis, a normal operator on a complex Hilbert space ''H'' is a continuous linear operator ''N'' : ''H'' → ''H'' that commutes with its hermitian adjoint ''N*'', that is: ''NN*'' = ''N*N''. Normal operat ...
s on finite dimensional inner product spaces. A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces.


Generalizations

Any of the axioms of an inner product may be weakened, yielding generalized notions. The generalizations that are closest to inner products occur where bilinearity and conjugate symmetry are retained, but positive-definiteness is weakened.


Degenerate inner products

If V is a vector space and \langle \,\cdot\,, \,\cdot\, \rangle a semi-definite sesquilinear form, then the function: \, x\, = \sqrt makes sense and satisfies all the properties of norm except that \, x\, = 0 does not imply x = 0 (such a functional is then called a semi-norm). We can produce an inner product space by considering the quotient W = V / \. The sesquilinear form \langle \,\cdot\,, \,\cdot\, \rangle factors through W. This construction is used in numerous contexts. The
Gelfand–Naimark–Segal construction In functional analysis, a discipline within mathematics, given a C*-algebra ''A'', the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic *-representations of ''A'' and certain linear functionals on ''A'' (called '' ...
is a particularly important example of the use of this technique. Another example is the representation of semi-definite kernels on arbitrary sets.


Nondegenerate conjugate symmetric forms

Alternatively, one may require that the pairing be a
nondegenerate form In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent definiti ...
, meaning that for all non-zero x \neq 0 there exists some y such that \langle x, y \rangle \neq 0, though y need not equal x; in other words, the induced map to the dual space V \to V^* is injective. This generalization is important in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
: a manifold whose tangent spaces have an inner product is a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
, while if this is related to nondegenerate conjugate symmetric form the manifold is a
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
. By
Sylvester's law of inertia Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real number, real quadratic form that remain invariant (mathematics), invariant under a change of basis. Namely, if ''A'' is the symme ...
, just as every inner product is similar to the dot product with positive weights on a set of vectors, every nondegenerate conjugate symmetric form is similar to the dot product with weights on a set of vectors, and the number of positive and negative weights are called respectively the positive index and negative index. Product of vectors in
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
is an example of indefinite inner product, although, technically speaking, it is not an inner product according to the standard definition above. Minkowski space has four
dimensions 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 coordina ...
and indices 3 and 1 (assignment of "+" and "−" to them differs depending on conventions). Purely algebraic statements (ones that do not use positivity) usually only rely on the nondegeneracy (the injective homomorphism V \to V^*) and thus hold more generally.


Related products

The term "inner product" is opposed to
outer product In linear algebra, the outer product of two coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An ea ...
, which is a slightly more general opposite. Simply, in coordinates, the inner product is the product of a 1 \times n with an n \times 1 vector, yielding a 1 \times 1 matrix (a scalar), while the outer product is the product of an m \times 1 vector with a 1 \times n covector, yielding an m \times n matrix. The outer product is defined for different dimensions, while the inner product requires the same dimension. If the dimensions are the same, then the inner product is the of the outer product (trace only being properly defined for square matrices). In an informal summary: "inner is horizontal times vertical and shrinks down, outer is vertical times horizontal and expands out". More abstractly, the outer product is the bilinear map W \times V^* \to \hom(V, W) sending a vector and a covector to a rank 1 linear transformation (
simple tensor In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties can be derived from their definitions, as linear maps or m ...
of type (1, 1)), while the inner product is the bilinear evaluation map V^* \times V \to F given by evaluating a covector on a vector; the order of the domain vector spaces here reflects the covector/vector distinction. The inner product and outer product should not be confused with the
interior product In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of d ...
and
exterior product In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of th ...
, which are instead operations on vector fields and
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s, or more generally on the
exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
. As a further complication, in
geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ge ...
the inner product and the (Grassmann) product are combined in the geometric product (the Clifford product in a
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
) – the inner product sends two vectors (1-vectors) to a scalar (a 0-vector), while the exterior product sends two vectors to a bivector (2-vector) – and in this context the exterior product is usually called the (alternatively, ). The inner product is more correctly called a product in this context, as the nondegenerate quadratic form in question need not be positive definite (need not be an inner product).


See also

* * * * * * * * *


Notes


References


Bibliography

* * * * * * * * * * * {{HilbertSpace Normed spaces Bilinear forms