In
linear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, a
linear functional or
linear form (also called a
oneformIn linear algebra, a oneform on a vector space is the same as a linear functional on the space. The usage of oneform in this context usually distinguishes the oneforms from higherdegree multilinear functionals on the space. For details, see linear functional.In differential geometry, a...
or
covector) is a linear map from a
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
to its field of
scalarIn linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
s. In
R^{n}In mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, if vectors are represented as
column vectors, then linear functionals are represented as
row vectors, and their action on vectors is given by the
dot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equallength sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
, or the matrix product with the
row vector on the left and the
column vector on the right. In general, if
V is a
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over a
fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
k, then a linear functional ƒ is a function from
V to
k, which is linear:
for all
for all
The set of all linear functionals from
V to
k, Hom
_{k}(
V,
k), is itself a vector space over
k. This space is called the
dual spaceIn mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finitedimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
of
V, or sometimes the
algebraic dual space, to distinguish it from the continuous dual space. It is often written
V^{*} or
when the field
k is understood.
Continuous linear functionals
If V is a
topological vector spaceIn mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
, the space of
continuousIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
linear functionals — the
continuous dual — is often simply called the dual space. If
V is a
Banach spaceIn mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm · such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
, then so is its (continuous) dual. To distinguish the ordinary dual space from the continuous dual space, the former is sometimes called the
algebraic dual. In finite dimensions, every linear functional is continuous, so the continuous dual is the same as the algebraic dual, although this is not true in infinite dimensions.
Linear functionals in R^{n}
Suppose that vectors in the real coordinate space
R^{n} are represented as column vectors
Then any linear functional can be written in these coordinates as a sum of the form:
This is just the matrix product of the row vector [
a_{1} ...
a_{n}] and the column vector
:
Integration
Linear functionals first appeared in
functional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limitrelated structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, the study of
vector spaces of functionsIn mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.Examples:...
. A typical example of a linear functional is
integrationIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
: the linear transformation defined by the
Riemann integralIn the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. The Riemann integral is unsuitable for many theoretical purposes...
is a linear functional from the vector space C[
a,
b] of continuous functions on the interval [
a,
b] to the real numbers. The linearity of
I(ƒ) follows from the standard facts about the integral:

Evaluation
Let
P_{n} denote the vector space of realvalued polynomials of degree ≤
n defined on an interval [
a,
b]. If
c ∈ [
a,
b], then let
ev_{c} :
P_{n} →
R be the
evaluation functional:
The mapping ƒ → ƒ(
c) is linear since
If
x_{0}, ...,
x_{n} are
n+1 distinct points in [
a,
b], then the evaluation functionals
ev_{xi},
i=0,1,...,
n form a basis of the dual space of
P_{n}. ( proves this last fact using Lagrange interpolation.)
Application to quadrature
The integration functional
I defined above defines a linear functional on the
subspaceThe concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
P_{n} of polynomials of degree ≤
n. If
x_{0}, …,
x_{n} are
n+1 distinct points in [
a,
b], then there are coefficients
a_{0}, …,
a_{n} for which
for all
ƒ ∈
P_{n}. This forms the foundation of the theory of numerical quadrature.
This follows from the fact that the linear functionals
ev_{xi} :
ƒ →
ƒ(
x_{i}) defined above form a basis of the dual space of
P_{n} .
Linear functionals in quantum mechanics
Linear functionals are particularly important in
quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
. Quantum mechanical systems are represented by
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the twodimensional Euclidean plane and threedimensional space to spaces with any finite or infinite number of dimensions...
s, which are antiisomorphic to their own dual spaces. A state of a quantum mechanical system can be identified with a linear functional. For more information see
braket notationBraket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
.
Distributions
In the theory of
generalized functionIn mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognized theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing physical phenomena such as point charges...
s, certain kinds of generalized functions called
distributionsIn mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
can be realized as linear functionals on spaces of test functions.
Properties
 Any linear functional is either trivial (equal to 0 everywhere) or surjective onto the scalar field. Indeed, this follows since the image of a vector subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
under a linear transformation is a subspace, so is the image of V under L. But the only subspaces (i.e., ksubspaces) of k are {0} and k itself.
 A linear functional is continuous if and only if its kernel is closed .
 Linear functionals with the same kernel are proportional.
 The absolute value of any linear functional is a seminorm on its vector space.
Dual vectors and bilinear forms
Every nondegenerate
bilinear form on a finitedimensional vector space
V gives rise to an
isomorphismIn abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
from
V to
V*. Specifically, denoting the bilinear form on
V by ( , ) (for instance in
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
(
v,
w) =
v•
w is the
dot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equallength sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
of
v and
w), then there is a natural isomorphism
given by

The inverse isomorphism is given by
where ƒ* is the unique element of
V for which for all
w ∈
V

The above defined vector
v* ∈
V* is said to be the
dual vector of
v ∈
V.
In an infinite dimensional
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the twodimensional Euclidean plane and threedimensional space to spaces with any finite or infinite number of dimensions...
, analogous results hold by the
Riesz representation theoremThere are several wellknown theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz. The Hilbert space representation theorem :...
. There is a mapping
V →
V* into the
continuous dual space V*. However, this mapping is antilinear rather than linear.
Visualizing linear functionals
In finite dimensions, a linear function can be visualized in terms of its
level setIn mathematics, a level set of a realvalued function f of n variables is a set of the formthat is, a set where the function takes on a given constant value c....
s. In three dimensions, the level sets of a linear functional are a family of mutually parallel planes; in higher dimensions, they are parallel
hyperplaneA hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an ndimensional space is a flat subset with dimension n − 1...
s. This method of visualizing linear functionals is sometimes introduced in
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
texts, such as .
Basis of the dual space in finite dimensions
Let the vector space
V have a basis
, not necessarily orthogonal. Then the
dual spaceIn mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finitedimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
V* has a basis
called the
dual basis defined by the special property that
Or, more succinctly,
where δ is the
Kronecker delta. Here the superscripts of the basis functionals are not exponents but are instead
contravariantIn multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis from one coordinate system to another. When one coordinate system is just a rotation of the other, this...
indices.
A linear functional
belonging to the dual space
can be expressed as a
linear combinationIn mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
of basis functionals, with coefficients ("components")
u_{i} ,
Then, applying the functional
to a basis vector
e_{j} yields
due to linearity of scalar multiples of functionals and pointwise linearity of sums of functionals. Then
that is
This last equation shows that an individual component of a linear functional can be extracted by applying the functional to a corresponding basis vector.
The dual basis and inner product
When the space
V carries an inner product, then it is possible to write explicitly a formula for the dual basis of a given basis. Let
V have (not necessarily orthogonal) basis
. In three dimensions (
n = 3), the dual basis can be written explicitly
for
i=1,2,3, where
is the
LeviCivita symbolThe LeviCivita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus...
and
the inner product (or
dot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equallength sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
) on
V.
In higher dimensions, this generalizes as follows
where
is the Hodge star operator.