In

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 ...

$x\; \backslash cdot\; y,$ where $x\; =\; a\; +\; i\; b\; \backslash in\; V\; =\; \backslash Complex$ is identified with the point $(a,\; b)\; \backslash in\; V\_\; =\; \backslash R^2$ (and similarly for $y$); thus the standard inner product $\backslash langle\; x,\; y\; \backslash rangle\; =\; x\; \backslash overline,$ on $\backslash Complex$ is an "extension" the dot product . Also, had $\backslash langle\; x,\; y\; \backslash rangle$ been instead defined to be the $\backslash langle\; x,\; y\; \backslash rangle\; =\; x\; y$ (rather than the usual $\backslash langle\; x,\; y\; \backslash rangle\; =\; x\; \backslash overline$) then its real part $\backslash langle\; x,\; y\; \backslash rangle\_$ would be the dot product; furthermore, without the complex conjugate, if $x\; \backslash in\; \backslash C$ but $x\; \backslash not\backslash in\; \backslash R$ then $\backslash langle\; x,\; x\; \backslash rangle\; =\; x\; x\; =\; x^2\; \backslash not\backslash in\; [0,\; \backslash infty)$ so the assignment $x\; \backslash mapsto\; \backslash sqrt$ 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 $\backslash langle\; x,\; y\; \backslash rangle\; =\; 0$ then $\backslash langle\; x,\; y\; \backslash rangle\_\; =\; 0,$ but the next example shows that the converse is in general true.
Given any $x\; \backslash in\; V,$ 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 $i\; =\; \backslash sqrt$ 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, $\backslash langle\; x,\; ix\; \backslash rangle\; =\; -i\; \backslash ,\; x\backslash ,\; ^2,$ whereas for the real inner product the value is always $\backslash langle\; x,\; ix\; \backslash rangle\_\; =\; 0.$
If $\backslash langle\; \backslash ,\backslash cdot,\; \backslash cdot\backslash ,\; \backslash rangle$ is a complex inner product and $A\; :\; V\; \backslash to\; V$ is a continuous linear operator that satisfies $\backslash langle\; x,\; A\; x\; \backslash rangle\; =\; 0$ for all $x\; \backslash in\; V,$ then $A\; =\; 0.$ This statement is no longer true if $\backslash langle\; \backslash ,\backslash cdot,\; \backslash cdot\backslash ,\; \backslash rangle$ is instead a real inner product, as this next example shows.
Suppose that $V\; =\; \backslash Complex$ has the inner product $\backslash langle\; x,\; y\; \backslash rangle\; :=\; x\; \backslash overline$ mentioned above. Then the map $A\; :\; V\; \backslash to\; V$ 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 $\backslash langle\; x,\; Ax\; \backslash rangle\_$ is just the dot product, $\backslash langle\; x,\; Ax\; \backslash rangle\_\; =\; 0$ for all vectors $x;$ nevertheless, this rotation map $A$ is certainly not identically $0.$ In contrast, using the complex inner product gives $\backslash langle\; x,\; Ax\; \backslash rangle\; =\; -i\; \backslash ,\; x\backslash ,\; ^2,$ which (as expected) is not identically zero.

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 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 $\backslash langle\; a,\; b\; \backslash rangle$. 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 ...

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 ...

(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 ...

. 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 ...

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 ...

, 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 $\backslash overline.$ 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 $\backslash overline,$ the inner product of $H$ is the restriction of that of $\backslash overline,$ 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 $\backslash overline$ 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, thefield
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 $\backslash R$ 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 ...

s $\backslash Complex$.
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
:$\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle\; :\; V\; \backslash times\; V\; \backslash to\; F$
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., $\backslash overline$ is the complex conjugation of $x$. For real values, $x\; =\; \backslash overline$ and conjugate symmetry is just symmetry.
*: $\backslash langle\; x,\; y\; \backslash rangle\; =\; \backslash overline$
* 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'': $\backslash langle\; x,by\; \backslash rangle\; =\; \backslash langle\; x,y\; \backslash rangle\; \backslash overline$. 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 $\backslash langle\; x,y\; \backslash rangle\; =\; x^*\; M\; y$. (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 $x$.)
*: $\backslash begin\; \backslash langle\; ax,\; y\; \backslash rangle\; \&=\; a\; \backslash langle\; x,\; y\; \backslash rangle\; \backslash \backslash \; \backslash langle\; x\; +\; y,\; z\; \backslash rangle\; \&=\; \backslash langle\; x,\; z\; \backslash rangle\; +\; \backslash langle\; y,\; z\; \backslash rangle\; \backslash end$
* Positive-definite:
*: $\backslash langle\; x,\; x\; \backslash rangle\; >\; 0,\backslash quad\; x\; \backslash in\; V\; \backslash setminus\; \backslash .$
If the positive-definite condition is replaced by merely requiring that $\backslash langle\; x,\; x\; \backslash rangle\; \backslash geq\; 0$ for all ''x'', then one obtains the definition of ''positive semi-definite Hermitian form''. A positive semi-definite Hermitian form $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ is an inner product if and only if for all ''x'', if $\backslash langle\; x,\; x\; \backslash rangle\; =\; 0$ then ''x = 0''.
Elementary properties

Positive-definiteness and linearity, respectively, ensure that: :$\backslash begin\; \backslash langle\; x,\; x\; \backslash rangle\; \&=\; 0\; \backslash Rightarrow\; x\; =\; \backslash mathbf\; \backslash \backslash \; \backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle\; \&=\; \backslash langle\; 0x,\; 0x\; \backslash rangle\; =\; 0\; \backslash langle\; x,\; 0x\; \backslash rangle\; =\; 0\; \backslash end$ Notice that conjugate symmetry implies that is real for all , since we have: : $\backslash langle\; x,\; x\; \backslash rangle\; =\; \backslash overline\; \backslash ,.$ Conjugate symmetry and linearity in the first variable imply :$\backslash begin\; \backslash langle\; x,\; a\; y\; \backslash rangle\; \&=\; \backslash overline\; =\; \backslash overline\; \backslash overline\; =\; \backslash overline\; \backslash langle\; x,\; y\; \backslash rangle\; \backslash \backslash \; \backslash langle\; x,\; y\; +\; z\; \backslash rangle\; \&=\; \backslash overline\; =\; \backslash overline\; +\; \backslash overline\; =\; \backslash langle\; x,\; y\; \backslash rangle\; +\; \backslash langle\; x,\; z\; \backslash rangle\; \backslash ,\; \backslash end$ that is, conjugate linearity in the second argument. So, an inner product is asesquilinear 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:
: $\backslash langle\; x\; +\; y,\; x\; +\; y\; \backslash rangle\; =\; \backslash langle\; x,\; x\; \backslash rangle\; +\; \backslash langle\; x,\; y\; \backslash rangle\; +\; \backslash langle\; y,\; x\; \backslash rangle\; +\; \backslash langle\; y,\; y\; \backslash rangle\; \backslash ,.$
These properties, constituents of the above linearity in the first and second argument:
:$\backslash begin\; \backslash langle\; x\; +\; y,\; z\; \backslash rangle\; \&=\; \backslash langle\; x,\; z\; \backslash rangle\; +\; \backslash langle\; y,\; z\; \backslash rangle\; \backslash ,,\; \backslash \backslash \; \backslash langle\; x,\; y\; +\; z\; \backslash rangle\; \&=\; \backslash langle\; x,\; y\backslash rangle\; +\; \backslash langle\; x,\; z\; \backslash rangle\; \backslash end$
are otherwise known as ''additivity''.
In the case of = $\backslash R$, 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,
:$\backslash begin\; \backslash langle\; x,\; y\; \backslash rangle\; \&=\; \backslash langle\; y,\; x\; \backslash rangle\; \backslash \backslash \; \backslash Rightarrow\; \backslash langle\; -x,\; x\; \backslash rangle\; \&=\; \backslash langle\; x,\; -x\; \backslash rangle\; \backslash ,,\; \backslash end$
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 ...

becomes:
: $\backslash langle\; x\; +\; y,\; x\; +\; y\; \backslash rangle\; =\; \backslash langle\; x,\; x\; \backslash rangle\; +\; 2\backslash langle\; x,\; y\; \backslash rangle\; +\; \backslash langle\; y,\; y\; \backslash rangle\; \backslash ,.$
Convention variant

Some authors, especially inphysics
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 ...

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 $\backslash R$ and $\backslash Complex.$ Thereal 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 $\backslash R$ are a vector space over $\backslash R$ that becomes a real inner product space when endowed with standard multiplication as its real inner product:
$$\backslash langle\; x,\; y\; \backslash rangle\; :=\; x\; y\; \backslash quad\; \backslash text\; x,\; y\; \backslash in\; \backslash R.$$
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 ...

s $\backslash Complex$ are a vector space over $\backslash Complex$ that becomes a complex inner product space when endowed with the complex inner product
$$\backslash langle\; x,\; y\; \backslash rangle\; :=\; x\; \backslash overline\; \backslash quad\; \backslash text\; x,\; y\; \backslash in\; \backslash Complex.$$
Unlike with the real numbers, the assignment $(x,\; y)\; \backslash mapsto\; x\; y$ does define a complex inner product on $\backslash Complex.$
Euclidean vector space

More generally, the real $n$-space $\backslash R^n$ with thedot 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 ...

.
$$\backslash left\backslash langle\; \backslash begin\; x\_1\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; x\_n\; \backslash end,\; \backslash begin\; y\_1\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; y\_n\; \backslash end\; \backslash right\backslash rangle\; =\; x^\backslash textsf\; y\; =\; \backslash sum\_^n\; x\_i\; y\_i\; =\; x\_1\; y\_1\; +\; \backslash cdots\; +\; x\_n\; y\_n,$$
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 ...

of $x.$
A function $\backslash langle\; \backslash ,\backslash cdot,\; \backslash cdot\backslash ,\; \backslash rangle\; :\; \backslash R^n\; \backslash times\; \backslash R^n\; \backslash to\; \backslash R$ is an inner product on $\backslash R^n$ 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 ...

$\backslash mathbf$ such that $\backslash langle\; x,\; y\; \backslash rangle\; =\; x^\; \backslash mathbf\; y$ for all $x,\; y\; \backslash in\; \backslash R^n.$ If $\backslash mathbf$ 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 ...

then $\backslash langle\; x,\; y\; \backslash rangle\; =\; x^\; \backslash mathbf\; y$ is the dot product. For another example, if $n\; =\; 2$ and $\backslash mathbf\; =\; \backslash begin\; a\; \&\; b\; \backslash \backslash \; b\; \&\; d\; \backslash end$ is positive-definite (which happens if and only if $\backslash det\; \backslash mathbf\; =\; a\; d\; -\; b^2\; >\; 0$ and one/both diagonal elements are positive) then for any $x\; :=\; \backslash left;\; href="/html/ALL/s/\_1,\_x\_2\backslash right.html"\; ;"title="\_1,\; x\_2\backslash right">\_1,\; x\_2\backslash right$
$$\backslash langle\; x,\; y\; \backslash rangle\; :=\; x^\; \backslash mathbf\; y\; =\; \backslash left;\; href="/html/ALL/s/\_1,\_x\_2\backslash right.html"\; ;"title="\_1,\; x\_2\backslash right">\_1,\; x\_2\backslash right$$
As mentioned earlier, every inner product on $\backslash R^2$ is of this form (where $b\; \backslash in\; \backslash R,\; a\; >\; 0$ and $d\; >\; 0$ satisfy $a\; d\; >\; b^2$).
Complex coordinate space

The general form of an inner product on $\backslash Complex^n$ is known as theHermitian 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
$$\backslash langle\; x,\; y\; \backslash rangle\; =\; y^\backslash dagger\; \backslash mathbf\; x\; =\; \backslash overline,$$
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 onHilbert 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
$$\backslash langle\; f,\; g\; \backslash rangle\; =\; \backslash int\_a^b\; f(t)\; \backslash overline\; \backslash ,\; \backslash mathrmt.$$
This space is not complete; consider for example, for the interval the sequence of continuous "step" functions, $\backslash \_k,$ defined by:
$$f\_k(t)\; =\; \backslash begin\; 0\; \&\; t\; \backslash in;\; href="/html/ALL/s/1,\_0.html"\; ;"title="1,\; 0">1,\; 0$$
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 realrandom 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
$$\backslash langle\; X,\; Y\; \backslash rangle\; =\; \backslash operatorname(XY)$$
is an inner product. In this case, $\backslash langle\; X,\; X\; \backslash rangle\; =\; 0$ if and only if $\backslash Pr\; (X\; =\; 0)\; =\; 1$ (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 $\backslash Pr$ 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 ...

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 $\backslash langle\; A,\; B\; \backslash rangle\; :=\; \backslash operatorname\backslash left(AB^\backslash right)$. 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, $$\backslash langle\; A,\; B\; \backslash rangle\; =\; \backslash operatorname\backslash left(AB^\backslash right)\; =\; \backslash overline\; =\; \backslash overline$$ Finally, since for $A$ nonzero, $\backslash langle\; A,\; A\backslash rangle\; =\; \backslash sum\_\; \backslash left,\; A\_\backslash ^2\; >\; 0$, 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 anondegenerate 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 $V\; \backslash to\; V^*$), 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 anorm
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
$$\backslash ,\; x\backslash ,\; =\; \backslash sqrt.$$
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
$$d(x,\; y)\; :=\; \backslash ,\; y\; -\; x\backslash ,\; =\; \backslash ,\; x\; -\; y\backslash ,\; .$$
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 $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ is an inner product on $V$ (so it is antilinear in its second argument). Thepolarization 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
$$\backslash operatorname\; \backslash langle\; x,\; y\; \backslash rangle\; =\; \backslash frac\; \backslash left(\backslash ,\; x\; +\; y\backslash ,\; ^2\; -\; \backslash ,\; x\; -\; y\backslash ,\; ^2\backslash right).$$
If $V$ is a real vector space then
$$\backslash langle\; x,\; y\; \backslash rangle\; =\; \backslash operatorname\; \backslash langle\; x,\; y\; \backslash rangle\; =\; \backslash frac\; \backslash left(\backslash ,\; x\; +\; y\backslash ,\; ^2\; -\; \backslash ,\; x\; -\; y\backslash ,\; ^2\backslash right)$$
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 $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ 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
:$\backslash begin\; \backslash langle\; x,\; \backslash \; y\; \backslash rangle\; \&=\; \backslash frac\; \backslash left(\backslash ,\; x\; +\; y\backslash ,\; ^2\; -\; \backslash ,\; x\; -\; y\backslash ,\; ^2\; +\; i\backslash ,\; x\; +\; iy\backslash ,\; ^2\; -\; i\backslash ,\; x\; -\; iy\backslash ,\; ^2\; \backslash right)\; \backslash \backslash \; \&=\; \backslash operatorname\; \backslash langle\; x,\; y\; \backslash rangle\; +\; i\; \backslash operatorname\; \backslash langle\; x,\; i\; y\; \backslash rangle.\; \backslash \backslash \; \backslash end$
The map defined by $\backslash langle\; x\; \backslash mid\; y\; \backslash rangle\; =\; \backslash langle\; y,\; x\; \backslash rangle$ for all $x,\; y\; \backslash in\; V$ satisfies the axioms of the inner product except that it is antilinear in its , rather than its second, argument. The real part of both $\backslash langle\; x\; \backslash mid\; y\; \backslash rangle$ and $\backslash langle\; x,\; y\; \backslash rangle$ are equal to $\backslash operatorname\; \backslash langle\; x,\; y\; \backslash rangle$ but the inner products differ in their complex part:
:$\backslash begin\; \backslash langle\; x\; \backslash mid\; y\; \backslash rangle\; \&=\; \backslash frac\; \backslash left(\backslash ,\; x\; +\; y\backslash ,\; ^2\; -\; \backslash ,\; x\; -\; y\backslash ,\; ^2\; -\; i\backslash ,\; x\; +\; iy\backslash ,\; ^2\; +\; i\backslash ,\; x\; -\; iy\backslash ,\; ^2\; \backslash right)\; \backslash \backslash \; \&=\; \backslash operatorname\; \backslash langle\; x,\; y\; \backslash rangle\; -\; i\; \backslash operatorname\; \backslash langle\; x,\; i\; y\; \backslash rangle.\; \backslash \backslash \; \backslash end$
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 $V\; :=\; \backslash Complex^n$ 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 $\backslash R^,$ with each $\backslash left(a\_1\; +\; i\; b\_1,\; \backslash ldots,\; a\_n\; +\; i\; b\_n\backslash right)\; \backslash in\; \backslash Complex^n$ identified with $\backslash left(a\_1,\; b\_1,\; \backslash ldots,\; a\_n,\; b\_n\backslash right)\; \backslash in\; \backslash R^$), 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 ...

$x\; \backslash ,\backslash cdot\backslash ,\; y\; =\; \backslash left(x\_1,\; \backslash ldots,\; x\_\backslash right)\; \backslash ,\; \backslash cdot\; \backslash ,\; \backslash left(y\_1,\; \backslash ldots,\; y\_\backslash right)\; :=\; x\_1\; y\_1\; +\; \backslash cdots\; +\; x\_\; y\_$ defines a real inner product on this space. The unique complex inner product $\backslash langle\; \backslash ,\backslash cdot,\; \backslash cdot\backslash ,\; \backslash rangle$ on $V\; =\; \backslash C^n$ induced by the dot product is the map that sends $c\; =\; \backslash left(c\_1,\; \backslash ldots,\; c\_n\backslash right),\; d\; =\; \backslash left(d\_1,\; \backslash ldots,\; d\_n\backslash right)\; \backslash in\; \backslash Complex^n$ to $\backslash langle\; c,\; d\; \backslash rangle\; :=\; c\_1\; \backslash overline\; +\; \backslash cdots\; +\; c\_n\; \backslash overline$ (because the real part of this map $\backslash langle\; \backslash ,\backslash cdot,\; \backslash cdot\backslash ,\; \backslash rangle$ 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 $\backslash langle\; x,\; y\; \backslash rangle$ is the map $\backslash langle\; x,\; y\; \backslash rangle\_\; =\; \backslash operatorname\; \backslash langle\; x,\; y\; \backslash rangle\; ~:~\; V\_\; \backslash times\; V\_\; \backslash to\; \backslash R,$ 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 $V\; =\; \backslash Complex$ with inner product $\backslash langle\; x,\; y\; \backslash rangle\; =\; x\; \backslash overline,$ where $V$ is a vector space over the field $\backslash Complex,$ then $V\_\; =\; \backslash R^2$ is a vector space over $\backslash R$ and $\backslash langle\; x,\; y\; \backslash rangle\_$ is the 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 $\backslash $ is orthonormal if $\backslash langle\; e\_i,\; e\_j\; \backslash rangle\; =\; 0$ for every $i\; \backslash neq\; j$ and $\backslash langle\; e\_i,\; e\_i\; \backslash rangle\; =\; \backslash ,\; e\_a\backslash ,\; ^2\; =\; 1$ 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 $$E\; =\; \backslash left\backslash \_$$ 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 $$\backslash left\backslash langle\; e\_,\; e\_\; \backslash right\backslash rangle\; =\; 0$$ if $a\; \backslash neq\; b$ and $\backslash langle\; e\_a,\; e\_a\; \backslash rangle\; =\; \backslash ,\; e\_a\backslash ,\; ^2\; =\; 1$ for all $a,\; b\; \backslash in\; A.$ Using an infinite-dimensional analog of the Gram-Schmidt process one may show: Theorem. Any separable inner product space has an orthonormal basis. Using theHausdorff 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 $\backslash left\backslash \_k$ an orthonormal basis of $V.$ Then the map
$$x\; \backslash mapsto\; \backslash bigl\backslash \_$$
is an isometric linear map $V\; \backslash mapsto\; \backslash ell^2$ 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 $\backslash ell^2$ 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 $C;\; href="/html/ALL/s/\backslash pi,\_\backslash pi.html"\; ;"title="\backslash pi,\; \backslash pi">\backslash pi,\; \backslash pi$ Then the sequence (indexed on set of all integers) of continuous functions
$$e\_k(t)\; =\; \backslash frac$$
is an orthonormal basis of the space $C;\; href="/html/ALL/s/\backslash pi,\_\backslash pi.html"\; ;"title="\backslash pi,\; \backslash pi">\backslash pi,\; \backslash pi$Operators on inner product spaces

Several types oflinear
Linearity is the property of a mathematical relationship (''function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out se ...

maps $A\; :\; V\; \backslash to\; W$ between inner product spaces $V$ and $W$ are of relevance:
* : $A\; :\; V\; \backslash to\; W$ is linear and continuous with respect to the metric defined above, or equivalently, $A$ is linear and the set of non-negative reals $\backslash ,$ where $x$ ranges over the closed unit ball of $V,$ is bounded.
* : $A\; :\; V\; \backslash to\; W$ is linear and $\backslash langle\; Ax,\; y\; \backslash rangle\; =\; \backslash langle\; x,\; Ay\; \backslash rangle$ for all $x,\; y\; \backslash in\; V.$
* : $A\; :\; V\; \backslash to\; W$ satisfies $\backslash ,\; A\; x\backslash ,\; =\; \backslash ,\; x\backslash ,$ for all $x\; \backslash in\; V.$ 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 $\backslash langle\; Ax,\; Ay\; \backslash rangle\; =\; \backslash langle\; x,\; y\; \backslash rangle$ for all $x,\; y\; \backslash in\; V.$ 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 ...

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 ...

).
* : $A\; :\; V\; \backslash to\; W$ 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 $\backslash langle\; \backslash ,\backslash cdot\backslash ,,\; \backslash ,\backslash cdot\backslash ,\; \backslash rangle$ a semi-definite sesquilinear form, then the function: $$\backslash ,\; x\backslash ,\; =\; \backslash sqrt$$ makes sense and satisfies all the properties of norm except that $\backslash ,\; x\backslash ,\; =\; 0$ 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 $W\; =\; V\; /\; \backslash .$ The sesquilinear form $\backslash langle\; \backslash ,\backslash cdot\backslash ,,\; \backslash ,\backslash cdot\backslash ,\; \backslash rangle$ 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 anondegenerate 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 $x\; \backslash neq\; 0$ there exists some $y$ such that $\backslash langle\; x,\; y\; \backslash rangle\; \backslash neq\; 0,$ though $y$ need not equal $x$; in other words, the induced map to the dual space $V\; \backslash to\; V^*$ 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 $V\; \backslash to\; V^*$) 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 $1\; \backslash times\; n$ with an $n\; \backslash times\; 1$ vector, yielding a $1\; \backslash times\; 1$ matrix (a scalar), while the outer product is the product of an $m\; \backslash times\; 1$ vector with a $1\; \backslash times\; n$ covector, yielding an $m\; \backslash times\; n$ 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 $W\; \backslash times\; V^*\; \backslash to\; \backslash hom(V,\; W)$ 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 $V^*\; \backslash times\; V\; \backslash to\; F$ 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