Linear form

In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the set of all linear functionals from to is itself a vector space over with addition and scalar multiplication defined pointwise. This space is called the dual space of , or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted , or, when the field formula_1 is understood, formula_2; other notations are also used, such as formula_3, formula_4 or formula_5 When vectors are represented by column vectors (as it is common when a basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by matrix products (with the row vector on the left).
Examples.
The "constant zero function," mapping every vector to zero, is trivially a linear functional. Every other linear functional (such as the ones below) is surjective (i.e. its range is all of ).
Linear functionals in R"n".
Suppose that vectors in the real coordinate space formula_6 are represented as column vectors
For each row vector ["a"1 ⋯ "a""n"] there is a linear functional "f" defined by
and each linear functional can be expressed in this form.
This can be interpreted as either the matrix product or the dot product of the row vector ["a"1 ⋯ "a""n"] and the column vector formula_9:
(Definite) Integration.
Linear functionals first appeared in functional analysis, the study of vector spaces of functions. A typical example of a linear functional is integration: the linear transformation defined by the Riemann integral
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 follows from the standard facts about the integral:
Evaluation.
Let "Pn" denote the vector space of real-valued polynomial functions of degree ≤"n" defined on an interval ["a", "b"]. If , then let formula_13 be the evaluation functional
The mapping "f" → "f"("c") is linear since
If "x"0, …, "xn" are distinct points in , then the evaluation functionals form a basis of the dual space of "Pn". ( proves this last fact using Lagrange interpolation.)
Non-example.
A function having the equation of a line with (e.g. ) is "not" a linear functional on formula_16, since it is not linear. It is, however, affine-linear. 
Visualization.
In finite dimensions, a linear functional can be visualized in terms of its level sets, the sets of vectors which map to a given value. In three dimensions, the level sets of a linear functional are a family of mutually parallel planes; in higher dimensions, they are parallel hyperplanes. This method of visualizing linear functionals is sometimes introduced in general relativity texts, such as "Gravitation" by .
Applications.
Application to quadrature.
If are distinct points in , then the linear functionals defined above form a basis of the dual space of , the space of polynomials of degree . The integration functional is also a linear functional on , and so can be expressed as a linear combination of these basis elements. In symbols, there are coefficients for which
for all . This forms the foundation of the theory of numerical quadrature.
In quantum mechanics.
Linear functionals are particularly important in quantum mechanics. Quantum mechanical systems are represented by Hilbert spaces, which are anti–isomorphic to their own dual spaces. A state of a quantum mechanical system can be identified with a linear functional. For more information see bra–ket notation.
Distributions.
In the theory of generalized functions, certain kinds of generalized functions called distributions can be realized as linear functionals on spaces of test functions.
Dual vectors and bilinear forms.
Every non-degenerate bilinear form on a finite-dimensional vector space "V" induces an isomorphism such that 
where the bilinear form on "V" is denoted (for instance, in Euclidean space is the dot product of "v" and "w").
The inverse isomorphism is , where "v" is the unique element of "V" such that
The above defined vector is said to be the dual vector of .
In an infinite dimensional Hilbert space, analogous results hold by the Riesz representation theorem. There is a mapping into the "continuous dual space" "V". 
Relationship to bases.
Basis of the dual space.
Let the vector space "V" have a basis formula_20, not necessarily orthogonal. Then the dual space "V*" has a basis formula_21 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 contravariant indices.
A linear functional formula_24 belonging to the dual space formula_25 can be expressed as a linear combination of basis functionals, with coefficients ("components") "ui", 
Then, applying the functional formula_24 to a basis vector e"j" yields
due to linearity of scalar multiples of functionals and pointwise linearity of sums of functionals. Then
So each component of a linear functional can be extracted by applying the functional to the 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 formula_30. In three dimensions (), the dual basis can be written explicitly
for "i" = 1, 2, 3, where "ε" is the Levi-Civita symbol and formula_32 the inner product (or dot product) on "V".
In higher dimensions, this generalizes as follows
where formula_34 is the Hodge star operator.
Over a ring.
Modules over a ring are generalizations of vector spaces, which removes the restriction that coefficients belong to a field. Given a module over a ring , a linear form on is a linear map from to , where the latter is considered as a module over itself. The space of linear forms is always denoted , whether is a field or not. It is an right module, if is a left module.
The existence of "enough" linear forms on a module is equivalent to projectivity. 
Change of field.
Any vector space over formula_35 is also a vector space over formula_16, endowed with a complex structure; that is, there exists a real vector subspace formula_37 such that we can (formally) write formula_38 as formula_16-vector spaces. Every formula_35-linear functional on is a formula_16-linear operator, but it is not an formula_16-linear "functional" on , because its range (namely, formula_35) is 2-dimensional over formula_16. (Conversely, a formula_16-linear functional has range too small to be a formula_35-linear functional as well.) 
However, every formula_35-linear functional uniquely determines an formula_16-linear functional on formula_37 by restriction. More surprisingly, this result can be reversed: every formula_16-linear functional on induces a canonical formula_35-linear functional , such that the real part of is : define 
The map is formula_16-linear (that is, and for all formula_53 and formula_54). Similarly, the inverse of the surjection formula_55 defined by is the map .
This relationship was discovered by Henry Löwig in 1934 (although it is usually credited to F. Murray), and can be generalized to arbitrary finite extensions of a field in the natural way.
In infinite dimensions.
Below, all vector spaces are over either the real numbers formula_16 or the complex numbers formula_35. 
If "V" is a topological vector space, the space of continuous linear functionals — the "continuous dual" — is often simply called the dual space. If "V" is a Banach space, 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 space". In finite dimensions, every linear functional is continuous, so the continuous dual is the same as the algebraic dual, but in infinite dimensions the continuous dual is a proper subspace of the algebraic dual.
A linear functional on a (not necessarily locally convex) topological vector space is continuous if and only if there exists a continuous seminorm on such that . 
Characterizing closed subspaces.
Continuous linear functionals have nice properties for analysis: a linear functional is continuous if and only if its kernel is closed, and a non-trivial continuous linear functional is an open map, even if the (topological) vector space is not complete. 
Hyperplanes and maximal subspaces.
A vector subspace of is called maximal if , but there are no vector subspaces satisfying . is maximal if and only if it is the kernel of some non-trivial linear functional on (i.e. for some non-trivial linear functional on ). A hyperplane in is a translate of a maximal vector subspace. By linearity, a subset of is a hyperplane if and only if there exists some non-trivial linear functional on such that .
Relationships between multiple linear functionals.
Any two linear functionals with the same kernel are proportional (i.e. scalar multiples of each other). 
This fact can be generalized to the following theorem. 
If is a non-trivial linear functional on with kernel , satisfies , and is a balanced subset of , then if and only if for all .
Hahn-Banach theorem.
Any (algebraic) linear functional on a vector subspace can be extended to the whole space; for example, the evaluation functionals described above can be extended to the vector space of polynomials on all of formula_16. However, this extension cannot always be done while keeping the linear functional continuous. The Hahn-Banach family of theorems gives conditions under which this extension can be done. For example, 
Equicontinuity of families of linear functionals.
Let be a topological vector space (TVS) with continuous dual space . 
For any subset of , the following are equivalent: 
If is an equicontinuous subset of then the following sets are also equicontinuous: 
the weak-* closure, the balanced hull, the convex hull, and the convex balanced hull. 
Moreover, Alaoglu's theorem implies that the weak-* closure of an equicontinuous subset of is weak-* compact (and thus that every equicontinuous subset weak-* relatively compact).