TheInfoList

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 has no generally ...
, a linear form (also known as a linear functional, a one-form, or a covector) is a
linear map 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 ...
from a vector space to its field of scalars (often, the
real number 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 ...
s or the
complex number 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 ...
s). 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 In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by consta ...
of , or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted , p. 19, §3.1 or, when the field $k$ is understood, $V^*$; other notations are also used, such as $V\text{'}$, $V^$ or $V^.$ When vectors are represented by
column vectorIn linear algebra, a column vector is a column of entries, for example, :\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end \,. Similarly, a row vector is a row of entries, p. 8 :\boldsymbol a = \begin a_1 & a_2 & \dots & a_n \end \,. Throu ...
s (as it is common when a basis is fixed), then linear functionals are represented as
row vectorIn linear algebra, a column vector is a column of entries, for example, :\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end \,. Similarly, a row vector is a row of entries, p. 8 :\boldsymbol a = \begin a_1 & a_2 & \dots & a_n \end \,. Throu ...
s, 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 $\mathbb^n$ are represented as column vectors :$\mathbf = \beginx_1\\ \vdots\\ x_n\end.$ For each row vector 'a''1 ⋯ ''a''''n''there is a linear functional ''f'' defined by :$f\left(\mathbf\right) = a_1x_1 + \cdots + a_n x_n,$ 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 $\mathbf$: :$f\left(\mathbf\right) = \begina_1 & \cdots & a_n\end \beginx_1\\ \vdots\\ x_n\end.$

(Definite) Integration

Linear functionals first appeared in
functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional an ...
, the study of vector spaces of functions. A typical example of a linear functional is integration: the linear transformation defined by the Riemann integral :$I\left(f\right) = \int_a^b f\left(x\right)\, dx$ 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: :$\begin I\left(f + g\right) &= \int_a^b\left[f\left(x\right) + g\left(x\right)\right]\, dx = \int_a^b f\left(x\right)\, dx + \int_a^b g\left(x\right)\, dx = I\left(f\right) + I\left(g\right) \\ I\left(\alpha f\right) &= \int_a^b \alpha f\left(x\right)\, dx = \alpha\int_a^b f\left(x\right)\, dx = \alpha I\left(f\right). \end$

Evaluation

Let ''Pn'' denote the vector space of real-valued polynomial functions of degree ≤''n'' defined on an interval [''a'', ''b'']. If , then let $\operatorname_c : P_n \rarr \R$ be the evaluation functional :$\operatorname_c f = f\left(c\right).$ The mapping ''f'' → ''f''(''c'') is linear since :$\begin \left(f + g\right)\left(c\right) &= f\left(c\right) + g\left(c\right) \\ \left(\alpha f\right)\left(c\right) &= \alpha f\left(c\right). \end$ If ''x''0, …, ''xn'' are distinct points in , then the evaluation functionals form a basis of a vector space, 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 $\mathbb$, since it is not Linear function, linear.For instance, . It is, however, Affine-linear function, 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 (book), ''Gravitation'' by .

Applications

If are distinct points in , then the linear functionals defined above form a basis of a vector space, 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 :$I\left(f\right) = a_0 f\left(x_0\right) + a_1 f\left(x_1\right) + \dots + a_n f\left(x_n\right)$ 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 antilinear, anti–linear isomorphism, 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 distribution (mathematics), 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 : $v^*\left(w\right) := \langle v, w\rangle \quad \forall w \in V ,$ 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 : $\langle v, w\rangle = v^*\left(w\right) \quad \forall w \in V .$ 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 $\mathbf_1, \mathbf_2,\dots,\mathbf_n$, not necessarily orthogonal. Then the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by consta ...
''V*'' has a basis $\tilde^1,\tilde^2,\dots,\tilde^n$ called the dual basis defined by the special property that :$\tilde^i \left(\mathbf e_j\right) = \begin 1 &\text\ i = j\\ 0 &\text\ i \neq j. \end$ Or, more succinctly, :$\tilde^i \left(\mathbf e_j\right) = \delta_$ where ''δ'' is the Kronecker delta. Here the superscripts of the basis functionals are not exponents but are instead covariance and contravariance of vectors, contravariant indices. A linear functional $\tilde$ belonging to the dual space $\tilde$ can be expressed as a linear combination of basis functionals, with coefficients ("components") ''ui'', :$\tilde = \sum_^n u_i \, \tilde^i.$ Then, applying the functional $\tilde$ to a basis vector e''j'' yields :$\tilde\left(\mathbf e_j\right) = \sum_^n \left\left(u_i \, \tilde^i\right\right) \mathbf e_j = \sum_i u_i \left\left[\tilde^i \left\left(\mathbf e_j\right\right)\right\right]$ due to linearity of scalar multiples of functionals and pointwise linearity of sums of functionals. Then :$\begin \tilde\left(_j\right) &= \sum_i u_i \left\left[\tilde^i \left\left(_j\right\right)\right\right] = \sum_i u_i _j \\ &= u_j. \end$ 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 $\mathbf_1,\dots, \mathbf_n$. In three dimensions (), the dual basis can be written explicitly :$\tilde^i\left(\mathbf\right) = \frac \left\langle \frac , \mathbf \right\rangle ,$ for ''i'' = 1, 2, 3, where ''ε'' is the Levi-Civita symbol and $\langle,\rangle$ the inner product (or dot product) on ''V''. In higher dimensions, this generalizes as follows :$\tilde^i\left(\mathbf\right) = \left\langle \frac, \mathbf \right\rangle ,$ where $\star$ is the Hodge star operator.

Over a ring

Module (mathematics), Modules over a ring (mathematics), 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 Projective module, projectivity.

Change of field

Any vector space over $\mathbb$ is also a vector space over $\mathbb$, endowed with a Linear complex structure, complex structure; that is, there exists a real vector subspace $X_\mathbb$ such that we can (formally) write $X = X_\mathbb \oplus X_\mathbbi$ as $\mathbb$-vector spaces. Every $\mathbb$-linear functional on is a Linear operator, $\mathbb$-linear operator, but it is not an $\mathbb$-linear ''functional'' on , because its range (namely, $\mathbb$) is 2-dimensional over $\mathbb$. (Conversely, a $\mathbb$-linear functional has range too small to be a $\mathbb$-linear functional as well.) However, every $\mathbb$-linear functional uniquely determines an $\mathbb$-linear functional on $X_\mathbb$ by Restriction (mathematics), restriction. More surprisingly, this result can be reversed: every $\mathbb$-linear functional on induces a canonical $\mathbb$-linear functional , such that the real part of is : define : for all . The map is $\mathbb$-linear (that is, and for all $r \isin \mathbb$ and $g, h \isin X_\mathbb$). Similarly, the inverse of the surjection $\text\left(X, \mathbb\right) \rarr \text\left(X, \mathbb\right)$ 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 field extension, finite extensions of a field in the natural way.

In infinite dimensions

Below, all vector spaces are over either the Real number, real numbers $\mathbb$ or the Complex number, complex numbers $\mathbb$. If ''V'' is a topological vector space, the space of Continuous function, continuous linear functionals — the ''Continuous dual space, 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 Real analysis, analysis: a linear functional is continuous if and only if its Kernel (linear operator), 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 set, 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 $\mathbb$. 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: # is Equicontinuity, equicontinuous; # is contained in the Polar set, polar of some neighborhood of in ; # the Polar set, (pre)polar of is a neighborhood of 0 in ; If is an equicontinuous subset of then the following sets are also equicontinuous: the Weak-* topology, weak-* closure, the Balanced set, balanced hull, the convex hull, and the Absolutely convex set, 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).