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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, an inner product space (or, rarely, a
Hausdorff
Hausdorff
pre-Hilbert space) is a
real vector space Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...
or a complex vector space with a
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 as ...
, 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 ...
, as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths,
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

angle
s, and
orthogonality In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

orthogonality
(zero inner product) of vectors. Inner product spaces generalize
Euclidean vector space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ...
s, in which the inner product is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
or ''scalar product'' of
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...

Cartesian coordinates
. Inner product spaces of infinite
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
are widely used in
functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...
. 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 grassl ...
of
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

Giuseppe Peano
, in 1898. An inner product naturally induces an associated
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
, (denoted , x, and , y, in the picture); so, every inner product space is a
normed vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
. If this normed space is also complete (that is, a
Banach space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
) then the inner product space is a
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of \overline, the inner product of H is the restriction of that of \overline, and H is
dense The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...
in \overline for the
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
defined by the norm.


Definition

In this article, 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 grassl ...
of
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 as ...
s denoted is either the field of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s \R or the field of
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s \Complex. Formally, an inner product space is a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
over the field together with an ''inner product'', i.e., with a map : \langle \cdot, \cdot \rangle : V \times V \to F that satisfies the following three properties for all vectors and all scalars : *
Conjugate Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ...
symmetry:A bar over an expression denotes complex conjugation; e.g., \overline is the complex conjugation of x. For real values, x = \overline and conjugate symmetry is just symmetry. *: \langle x, y \rangle = \overline *
Linearity Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to ''Proportionality (math ...
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 still used in some old-school math communities. However, all of engineering and computer science, and most of physics and modern mathematics now define the inner product to be ''linear in the second argument'' and ''conjugate-linear in the first argument'' because this is more compatible with several other conventions in mathematics. Notably, for any inner product, there is some
hermitian{{short description, Wikipedia list article Numerous things are named after the French mathematician Charles Hermite Charles Hermite () FRS FRSE Fellowship of the Royal Society of Edinburgh (FRSE) is an award granted to individuals that the R ...
, positive-definite matrix M such that \langle x,y \rangle = x^* M y . (Here, x^* is the
conjugate transpose In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of x.)
*: \begin \langle ax, y \rangle &= a \langle x, y \rangle \\ \langle x + y, z \rangle &= \langle x, z \rangle + \langle y, z \rangle \end * Positive-definite: *: \langle x, x \rangle > 0,\quad x \in V \setminus \. If the positive-definite condition is replaced by merely requiring that \langle x, x \rangle \geq 0 for all ''x'', 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''.


Elementary properties

Positive-definiteness and linearity, respectively, ensure that: :\begin \langle x, x \rangle &= 0 \Rightarrow x = \mathbf \\ \langle \mathbf, \mathbf \rangle &= \langle 0x, 0x \rangle = 0 \langle x, 0x \rangle = 0 \end Notice that conjugate symmetry implies that is real for all , since we have: : \langle x, x \rangle = \overline \,. Conjugate symmetry and linearity in the first variable imply :\begin \langle x, a y \rangle &= \overline = \overline \overline = \overline \langle x, y \rangle \\ \langle x, y + z \rangle &= \overline = \overline + \overline = \langle x, y \rangle + \langle x, z \rangle \, \end that is, conjugate linearity in the second argument. So, an inner product is a
sesquilinear form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. Conjugate symmetry is also called Hermitian symmetry, and a conjugate-symmetric sesquilinear form is called a ''Hermitian form''. While the above axioms are more mathematically economical, a compact verbal definition of an inner product is a ''positive-definite Hermitian form''. This important generalization of the familiar square expansion follows: : \langle x + y, x + y \rangle = \langle x, x \rangle + \langle x, y \rangle + \langle y, x \rangle + \langle y, y \rangle \,. These properties, constituents of the above linearity in the first and second argument: :\begin \langle x + y, z \rangle &= \langle x, z \rangle + \langle y, z \rangle \,, \\ \langle x, y + z \rangle &= \langle x, y\rangle + \langle x, z \rangle \end are otherwise known as ''additivity''. In the case of = \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''. That is, :\begin \langle x, y \rangle &= \langle y, x \rangle \\ \Rightarrow \langle -x, x \rangle &= \langle x, -x \rangle \,, \end and the
binomial expansion In elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ...

binomial expansion
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 Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
and
matrix algebra In abstract algebra, a matrix ring is a set of matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of ...
, prefer to define the inner product and the sesquilinear form with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second.


Some examples


Real and complex numbers

Among the simplest examples of inner product spaces are \R and \Complex. The
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s \R are a vector space over \R that becomes a real inner product space when endowed with standard multiplication as its real 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 contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s \Complex are a vector space over \Complex that becomes a complex inner product space when endowed with the complex 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'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
is an inner product space, an example of a
Euclidean vector space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ...
. \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 Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...

transpose
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 Greek συμμετρία ''symmetria'' "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 pre ...
positive-definite matrix In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
\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'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

identity matrix
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 map, linear in each of its arguments, but a sesquilinear fo ...
and is given by \langle x, y \rangle = y^\dagger \mathbf x = \overline, where M is any
Hermitian{{short description, Wikipedia list article Numerous things are named after the French mathematician Charles Hermite Charles Hermite () FRS FRSE Fellowship of the Royal Society of Edinburgh (FRSE) is an award granted to individuals that the R ...
positive-definite matrix In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
and y^ is the
conjugate transpose In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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, energy ...
of the two vectors, with positive
scale factor A scale factor is usually a decimal which scales Scale or scales may refer to: Mathematics * Scale (descriptive set theory)In the mathematical discipline of descriptive set theory, a scale is a certain kind of object defined on a set (mathemat ...
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 The mathematics, mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of linear algebra, vector algebra and calculus from the two-dimensional plane (geometry), Eucl ...
has several examples of inner product spaces, wherein the metric induced by the inner product yields a
complete metric space In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathema ...
. An example of an inner product space which induces an incomplete metric is the space C(
, b The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
of continuous complex valued functions f and g on the interval
, b The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
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 (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
s X and Y, the
expected value In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and space ...
of their product \langle X, Y \rangle = \operatorname(XY) is an inner product. In this case, \langle X, X \rangle = 0 if and only if \Pr (X = 0) = 1 (that is, X = 0
almost surely In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...
), where \Pr denotes the
probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

probability
of the event. This definition of expectation as inner product can be extended to
random vector In probability Probability is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculu ...
s as well.


Complex matrices

The inner product for complex square matrices of the same size is the Frobenius inner product \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 definition when ' ...
(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

Every inner product space induces a
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
, called its , that is defined by \, x\, = \sqrt. With this norm, every inner product space becomes a
normed vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
. As for every normed vector space, an inner product space is a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, for the distance defined by d(x, y) := \, y - x\, = \, x - y\, . The axioms of the inner product guarantee that the map above forms a norm, which will have 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathem ...
shows that the
real part In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...
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 a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...
(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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathem ...
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. There is thus a one-to-one correspondence between complex inner products and real inner products. 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'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
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 a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...
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 In mathematics, symmetrization is a process that converts any Function (mathematics), function in ''n'' variables to a symmetric function in ''n'' variables. Similarly, anti-symmetrization converts any function in ''n'' variables into an antisymmetr ...
. 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'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
, in which an arbitrary orthonormal basis plays the role of the sequence of
trigonometric polynomialIn the mathematical subfields of numerical analysis (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296... Numeri ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
). 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
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linear
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
). For inner product spaces, the
polarization identity In linear algebra, a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathem ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. The Mazur–Ulam theorem establishes that every surjective isometry between two normed spaces is an
affine transformation In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

morphism
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 In linear algebra, t ...
). * : A : V \to W is an isometry which is
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(and hence
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
). Isometrical isomorphisms are also known as unitary operators (compare with
unitary matrix In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
). 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 provides a canonical form for symmetric, unitary and more generally
normal operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 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 definition when ' ...
, 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: a manifold whose tangent spaces have an inner product is a Riemannian manifold, while if this is related to nondegenerate conjugate symmetric form the manifold is a pseudo-Riemannian manifold. By Sylvester's law of inertia, 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 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 Dimension (mathematics), dimensions and indices 3 and 1 (assignment of Sign (mathematics), "+" and "−" to them Sign convention#Metric signature, 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, 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 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 and exterior product, which are instead operations on vector fields and differential forms, or more generally on the exterior algebra. As a further complication, in geometric algebra the inner product and the (Grassmann) product are combined in the geometric product (the Clifford product in a Clifford algebra) – 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