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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a linear form (also known as a linear functional, a
one-form In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ea ...
, or a covector) is a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
from a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
to its
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 grass ...
of scalars (often, the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s or the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
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 In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
. 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 const ...
of , or sometimes the algebraic dual space, when a
topological 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 const ...
is also considered. It is often denoted , p. 19, §3.1 or, when the field is understood, V^*; other notations are also used, such as V', V^ or V^. When vectors are represented by
column vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
s (as is common when a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
is fixed), then linear functionals are represented as
row vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
s, and their values on specific vectors are given by
matrix product In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
s (with the row vector on the left).


Examples

* The constant
zero function 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
, mapping every vector to zero, is trivially a linear functional. * Indexing into a vector: The second element of a three-vector is given by the one-form , 1, 0 That is, the second element of , y, z/math> is , 1, 0\cdot , y, z= y. *
Mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arithme ...
: The mean element of an n-vector is given by the one-form \left /n, 1/n, \ldots, 1/n\right That is, \operatorname(v) = \left /n, 1/n, \ldots, 1/n\right\cdot v. * Sampling: Sampling with a kernel can be considered a one-form, where the one-form is the kernel shifted to the appropriate location. *
Net present value The net present value (NPV) or net present worth (NPW) applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the cash flow. It also depends on the discount ...
of a net
cash flow A cash flow is a real or virtual movement of money: *a cash flow in its narrow sense is a payment (in a currency), especially from one central bank account to another; the term 'cash flow' is mostly used to describe payments that are expected ...
, R(t), is given by the one-form w(t) = (1 + i)^ where i is the discount rate. That is, \mathrm(R(t)) = \langle w, R\rangle = \int_^\infty \frac\,dt. Every other linear functional (such as the ones below) is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
(that is, its range is all of ).


Linear functionals in R''n''

Suppose that vectors in the real coordinate space \R^n are represented as column vectors \mathbf = \beginx_1\\ \vdots\\ x_n\end. For each row vector \mathbf = \begina_1 & \cdots & a_n\end there is a linear functional f_ defined by f_(\mathbf) = a_1 x_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 \mathbf and the column vector \mathbf: f_(\mathbf) = \mathbf \cdot \mathbf = \begina_1 & \cdots & a_n\end \beginx_1\\ \vdots\\ x_n\end.


Trace of a square matrix

The
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' ...
\operatorname (A) of a square matrix A is the sum of all elements on its
main diagonal In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matri ...
. Matrices can be multiplied by scalars and two matrices of the same dimension can be added together; these operations make a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
from the set of all n \times n matrices. The trace is a linear functional on this space because \operatorname (s A) = s \operatorname (A) and \operatorname (A + B) = \operatorname (A) + \operatorname (B) for all scalars s and all n \times n matrices A \text B.


(Definite) Integration

Linear functionals first appeared in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, the study of vector spaces of functions. A typical example of a linear functional is
integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...
: the linear transformation defined by the
Riemann integral In 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. It was presented to the faculty at the University of Göt ...
I(f) = \int_a^b f(x)\, dx is a linear functional from the vector space C , b/math> of continuous functions on the interval , b/math> to the real numbers. The linearity of I follows from the standard facts about the integral: \begin I(f + g) &= \int_a^b
(x) + g(x) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, m ...
, dx = \int_a^b f(x)\, dx + \int_a^b g(x)\, dx = I(f) + I(g) \\ I(\alpha f) &= \int_a^b \alpha f(x)\, dx = \alpha\int_a^b f(x)\, dx = \alpha I(f). \end


Evaluation

Let P_n denote the vector space of real-valued polynomial functions of degree \leq n defined on an interval , b If c \in , b then let \operatorname_c : P_n \to \R be the evaluation functional \operatorname_c f = f(c). The mapping f \mapsto f(c) is linear since \begin (f + g)(c) &= f(c) + g(c) \\ (\alpha f)(c) &= \alpha f(c). \end If x_0, \ldots, x_n are n + 1 distinct points in , b then the evaluation functionals \operatorname_, i = 0, \ldots, n form a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
of the dual space of P_n ( proves this last fact using
Lagrange interpolation In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs (x_j, y_j) with 0 \leq j \leq k, the x_j are called ''nodes'' an ...
).


Non-example

A function f having the
equation of a line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segment ...
f(x) = a + r x with a \neq 0 (for example, f(x) = 1 + 2 x) is a linear functional on \R, since it is not
linear Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
.For instance, f(1 + 1) = a + 2 r \neq 2 a + 2 r = f(1) + f(1). It is, however, affine-linear.


Visualization

In finite dimensions, a linear functional can be visualized in terms of its
level set In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is calle ...
s, 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
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
s. This method of visualizing linear functionals is sometimes introduced in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
texts, such as ''Gravitation'' by .


Applications


Application to quadrature

If x_0, \ldots, x_n are n + 1 distinct points in , then the linear functionals \operatorname_ : f \mapsto f\left(x_i\right) defined above form a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
of the dual space of , the space of polynomials of degree \leq n. 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 a_0, \ldots, a_n for which I(f) = a_0 f(x_0) + a_1 f(x_1) + \dots + a_n f(x_n) for all f \in P_n. This forms the foundation of the theory of
numerical quadrature In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...
.


In quantum mechanics

Linear functionals are particularly important in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. Quantum mechanical systems are represented by
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s, which are
anti Anti may refer to: *Anti-, a prefix meaning "against" *Änti, or Antaeus, a half-giant in Greek and Berber mythology *A false reading of ''Nemty'', the name of the ferryman who carried Isis to Set's island in Egyptian mythology * Áńt’į, or ...
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
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 function In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
s, certain kinds of generalized functions called distributions can be realized as linear functionals on spaces of
test function Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
s.


Dual vectors and bilinear forms

Every non-degenerate
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
on a finite-dimensional vector space induces an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
such that v^*(w) := \langle v, w\rangle \quad \forall w \in V , where the bilinear form on is denoted \langle \,\cdot\, , \,\cdot\, \rangle (for instance, in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, \langle v, w \rangle = v \cdot w is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of and ). The inverse isomorphism is , where is the unique element of such that \langle v, w\rangle = v^*(w) for all w \in V. The above defined vector is said to be the dual vector of v \in V. In an infinite dimensional
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, analogous results hold by the
Riesz representation theorem :''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, sometimes called the R ...
. There is a mapping from into its ''V''.


Relationship to bases


Basis of the dual space

Let the vector space have a basis \mathbf_1, \mathbf_2,\dots,\mathbf_n, not necessarily
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
. 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 const ...
V^* has a basis \tilde^1,\tilde^2,\dots,\tilde^n called the
dual basis In linear algebra, given a vector space ''V'' with a basis ''B'' of vectors indexed by an index set ''I'' (the cardinality of ''I'' is the dimension of ''V''), the dual set of ''B'' is a set ''B''∗ of vectors in the dual space ''V''∗ with the ...
defined by the special property that \tilde^i (\mathbf e_j) = \begin 1 &\text\ i = j\\ 0 &\text\ i \neq j. \end Or, more succinctly, \tilde^i (\mathbf e_j) = \delta_ where ''δ'' is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
. Here the superscripts of the basis functionals are not exponents but are instead 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") , \tilde = \sum_^n u_i \, \tilde^i. Then, applying the functional \tilde to a basis vector \mathbf_j yields \tilde(\mathbf e_j) = \sum_^n \left(u_i \, \tilde^i\right) \mathbf e_j = \sum_i u_i \left tilde^i \left(\mathbf e_j\right)\right due to linearity of scalar multiples of functionals and pointwise linearity of sums of functionals. Then \begin \tilde(_j) &= \sum_i u_i \left tilde^i \left(_j\right)\right\\& = \sum_i u_i _ \\ &= 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 carries an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, then it is possible to write explicitly a formula for the dual basis of a given basis. Let have (not necessarily orthogonal) basis \mathbf_1,\dots, \mathbf_n. In three dimensions (), the dual basis can be written explicitly \tilde^i(\mathbf) = \frac \left\langle \frac , \mathbf \right\rangle , for i = 1, 2, 3, where ''ε'' is the
Levi-Civita symbol In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the parity of a permutation, sign of a permutation of the n ...
and \langle \cdot , \cdot \rangle the inner product (or
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
) on . In higher dimensions, this generalizes as follows \tilde^i(\mathbf) = \left\langle \frac, \mathbf \right\rangle , where \star is the
Hodge star operator In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
.


Over a ring

Modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
over a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
are generalizations of vector spaces, which removes the restriction that coefficients belong to a
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 grass ...
. 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 In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
, if is a left module. The existence of "enough" linear forms on a module is equivalent to
projectivity In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
.


Change of field

Suppose that X is a vector space over \Complex. Restricting scalar multiplication to \R gives rise to a real vector space X_ called the of X. Any vector space X over \Complex is also a vector space over \R, endowed with a complex structure; that is, there exists a real
vector subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
X_ such that we can (formally) write X = X_ \oplus X_i as \R-vector spaces. Real versus complex linear functionals Every linear functional on X is complex-valued while every linear functional on X_ is real-valued. If \dim X \neq 0 then a linear functional on either one of X or X_ is non-trivial (meaning not identically 0) if and only if it is surjective (because if \varphi(x) \neq 0 then for any scalar s, \varphi\left((s/\varphi(x)) x\right) = s), where the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of a linear functional on X is \C while the image of a linear functional on X_ is \R. Consequently, the only function on X that is both a linear functional on X and a linear function on X_ is the trivial functional; in other words, X^ \cap X_^ = \, where \,^ denotes the space's
algebraic 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 con ...
. However, every \Complex-linear functional on X is an \R-linear (meaning that it is
additive Additive may refer to: Mathematics * Additive function, a function in number theory * Additive map, a function that preserves the addition operation * Additive set-functionn see Sigma additivity * Additive category, a preadditive category with f ...
and homogeneous over \R), but unless it is identically 0, it is not an \R-linear on X because its range (which is \Complex) is 2-dimensional over \R. Conversely, a non-zero \R-linear functional has range too small to be a \Complex-linear functional as well. Real and imaginary parts If \varphi \in X^ then denote its
real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
by \varphi_ := \operatorname \varphi and its
imaginary part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
by \varphi_i := \operatorname \varphi. Then \varphi_ : X \to \R and \varphi_i : X \to \R are linear functionals on X_ and \varphi = \varphi_ + i \varphi_i. The fact that z = \operatorname z - i \operatorname (i z) = \operatorname (i z) + i \operatorname z for all z \in \Complex implies that for all x \in X, \begin\varphi(x) &= \varphi_(x) - i \varphi_(i x) \\ &= \varphi_i(i x) + i \varphi_i(x)\\ \end and consequently, that \varphi_i(x) = - \varphi_(i x) and \varphi_(x) = \varphi_i(ix). The assignment \varphi \mapsto \varphi_ defines a
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
\R-linear operator X^ \to X_^ whose inverse is the map L_ : X_^ \to X^ defined by the assignment g \mapsto L_g that sends g : X_ \to \R to the linear functional L_g : X \to \Complex defined by L_g(x) := g(x) - i g(ix) \quad \text x \in X. The real part of L_g is g and the bijection L_ : X_^ \to X^ is an \R-linear operator, meaning that L_ = L_g + L_h and L_ = r L_g for all r \in \R and g, h \in X_\R^. Similarly for the imaginary part, the assignment \varphi \mapsto \varphi_i induces an \R-linear bijection X^ \to X_^ whose inverse is the map X_^ \to X^ defined by sending I \in X_^ to the linear functional on X defined by x \mapsto I(i x) + i I(x). 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. It has many important consequences, some of which will now be described. Properties and relationships Suppose \varphi : X \to \Complex is a linear functional on X with real part \varphi_ := \operatorname \varphi and imaginary part \varphi_i := \operatorname \varphi. Then \varphi = 0 if and only if \varphi_ = 0 if and only if \varphi_i = 0. Assume that X is a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
. Then \varphi is continuous if and only if its real part \varphi_ is continuous, if and only if \varphi's imaginary part \varphi_i is continuous. That is, either all three of \varphi, \varphi_, and \varphi_i are continuous or none are continuous. This remains true if the word "continuous" is replaced with the word " bounded". In particular, \varphi \in X^ if and only if \varphi_ \in X_^ where the prime denotes the space's
continuous 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 cons ...
. Let B \subseteq X. If u B \subseteq B for all scalars u \in \Complex of
unit length Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (al ...
(meaning , u, = 1) thenIt is true if B = \varnothing so assume otherwise. Since \left, \operatorname z\ \leq , z, for all scalars z \in \Complex, it follows that \sup_ \left, \varphi_(x)\ \leq \sup_ , \varphi(x), . If b \in B then let r_b \geq 0 and u_b \in \Complex be such that \left, u_b\ = 1 and \varphi(b) = r_b u_b, where if r_b = 0 then take u_b := 1.Then , \varphi(b), = r_b and because \varphi\left(\frac b\right) = r_b is a real number, \varphi_\left(\frac b\right) = \varphi\left(\frac b\right) = r_b. By assumption \frac b \in B so , \varphi(b), = r_b \leq \sup_ \left, \varphi_(x)\. Since b \in B was arbitrary, it follows that \sup_ , \varphi(x), \leq \sup_ \left, \varphi_(x)\. \blacksquare \sup_ , \varphi(b), = \sup_ \left, \varphi_(b)\. Similarly, if \varphi_i := \operatorname \varphi : X \to \R denotes the complex part of \varphi then i B \subseteq B implies \sup_ \left, \varphi_(b)\ = \sup_ \left, \varphi_i(b)\. If X is a normed space with norm \, \cdot\, and if B = \ is the closed unit ball then the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
s above are the
operator norm In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Introd ...
s (defined in the usual way) of \varphi, \varphi_, and \varphi_i so that \, \varphi\, = \left\, \varphi_\right\, = \left\, \varphi_i \right\, . This conclusion extends to the analogous statement for polars of
balanced set In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, \ ...
s in general
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s. * If X is a complex
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
with a (complex)
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
\langle \,\cdot\,, \,\cdot\, \rangle that is
antilinear In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if \begin f(x + y) &= f(x) + f(y) && \qquad \text \\ f(s x) &= \overline f(x) && \qquad \text \\ \end hold for all vectors x, y \ ...
in its first coordinate (and linear in the second) then X_ becomes a real Hilbert space when endowed with the real part of \langle \,\cdot\,, \,\cdot\, \rangle. Explicitly, this real inner product on X_ is defined by \langle x , y \rangle_ := \operatorname \langle x , y \rangle for all x, y \in X and it induces the same norm on X as \langle \,\cdot\,, \,\cdot\, \rangle because \sqrt = \sqrt for all vectors x. Applying the
Riesz representation theorem :''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, sometimes called the R ...
to \varphi \in X^ (resp. to \varphi_ \in X_^) guarantees the existence of a unique vector f_ \in X (resp. f_ \in X_) such that \varphi(x) = \left\langle f_ , \, x \right\rangle (resp. \varphi_(x) = \left\langle f_ , \, x \right\rangle_) for all vectors x. The theorem also guarantees that \left\, f_\right\, = \, \varphi\, _ and \left\, f_\right\, = \left\, \varphi_\right\, _. It is readily verified that f_ = f_. Now \left\, f_\right\, = \left\, f_\right\, and the previous equalities imply that \, \varphi\, _ = \left\, \varphi_\right\, _, which is the same conclusion that was reached above.


In infinite dimensions

Below, all
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s are over either the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s \R or the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s \Complex. If V is a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
, the space of
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
linear functionals — the — is often simply called the dual space. If V is a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, then so is its (continuous) dual. To distinguish the ordinary dual space from the continuous dual space, the former is sometimes called the . 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 In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
is continuous if and only if there exists a continuous seminorm on such that , f, \leq p.


Characterizing closed subspaces

Continuous linear functionals have nice properties for
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
: a linear functional is continuous if and only if its
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
is closed, and a non-trivial continuous linear functional is an
open map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, a ...
, even if the (topological) vector space is not complete.


Hyperplanes and maximal subspaces

A vector subspace M of X is called maximal if M \subsetneq X (meaning M \subseteq X and M \neq X) and does not exist a vector subspace N of X such that M \subsetneq N \subsetneq X. A vector subspace M of X is maximal if and only if it is the kernel of some non-trivial linear functional on X (that is, M = \ker f for some linear functional f on X that is not identically ). An affine hyperplane in X is a translate of a maximal vector subspace. By linearity, a subset H of X is a affine hyperplane if and only if there exists some non-trivial linear functional f on X such that H = f^(1) = \. If f is a linear functional and s \neq 0 is a scalar then f^(s) = s \left(f^(1)\right) = \left(\frac f\right)^(1). This equality can be used to relate different level sets of f. Moreover, if f \neq 0 then the kernel of f can be reconstructed from the affine hyperplane H := f^(1) by \ker f = H - H.


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 , x \in X satisfies f(x) = 1, and is a
balanced In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ci ...
subset of , then N \cap (x + U) = \varnothing if and only if , f(u), < 1 for all u \in U.


Hahn–Banach theorem

Any (algebraic) linear functional on a
vector subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
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 \R. 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 In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
(TVS) with
continuous 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 cons ...
X'. For any subset of X', the following are equivalent: # is
equicontinuous In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable fa ...
; # is contained in the
polar Polar may refer to: Geography Polar may refer to: * Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates * Polar climate, the c ...
of some neighborhood of 0 in ; # the (pre)polar of is a neighborhood of 0 in ; If is an equicontinuous subset of X' then the following sets are also equicontinuous: the weak-* closure, the
balanced hull In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, ...
, the
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
, and the
convex balanced hull In mathematics, a subset ''C'' of a Real number, real or Complex number, complex vector space is said to be absolutely convex or disked if it is Convex set, convex and Balanced set, balanced (some people use the term "circled" instead of "balanced") ...
. Moreover, Alaoglu's theorem implies that the weak-* closure of an equicontinuous subset of X' is weak-* compact (and thus that every equicontinuous subset weak-* relatively compact).


See also

* * * * *


Notes


Footnotes


Proofs


References


Bibliography

* * * * * * * * * * * * * * * {{TopologicalVectorSpaces Functional analysis Linear algebra Linear operators