Dual Vector Space
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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 ...
, any
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 ...
''V'' has a corresponding dual vector space (or just dual space for short) consisting of all
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 s ...
s on ''V'', together with the vector space structure of
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 ...
addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for 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 ...
, there is a subspace of the dual space, corresponding to continuous linear functionals, called the ''continuous dual space''. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
analysis with
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe
measures Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Measu ...
, distributions, and
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. Consequently, the dual space is an important concept 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 ...
. Early terms for ''dual'' include ''polarer Raum'' ahn 1927 ''espace conjugué'', ''adjoint space'' laoglu 1940 and ''transponierter Raum'' chauder 1930and anach 1932 The term ''dual'' is due to Bourbaki 1938.


Algebraic dual space

Given any
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 ...
V over 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 ...
F, the (algebraic) dual space V^ (alternatively denoted by V^ p. 19, §3.1 or V')For V^ used in this way, see '' An Introduction to Manifolds'' (). This notation is sometimes used when (\cdot)^* is reserved for some other meaning. For instance, in the above text, F^* is frequently used to denote the codifferential of ''F'', so that F^* \omega represents the pullback of the form \omega. uses V' to denote the algebraic dual of ''V''. However, other authors use V' for the continuous dual, while reserving V^* for the algebraic dual (). is defined as the set of all
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 ...
s ''\varphi: V \to F'' (
linear functional 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 s ...
s). Since linear maps are vector space
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
s, the dual space may be denoted \hom (V, F). p. 19, §3.1 The dual space V^* itself becomes a vector space over ''F'' when equipped with an addition and scalar multiplication satisfying: : \begin (\varphi + \psi)(x) &= \varphi(x) + \psi(x) \\ (a \varphi)(x) &= a \left(\varphi(x)\right) \end for all \varphi, \psi \in V^*, ''x \in V'', and a \in F. Elements of the algebraic dual space V^* are sometimes called covectors or
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 ...
s. The pairing of a functional ''\varphi'' in the dual space V^* and an element ''x'' of ''V'' is sometimes denoted by a bracket: ''\varphi (x) = , \varphi/math>'' or ''\varphi (x) = \langle x, \varphi \rangle''. This pairing defines a nondegenerate
bilinear mapping In mathematics, a bilinear map is a Function (mathematics), function combining elements of two vector spaces to yield an element of a third vector space, and is Linear map, linear in each of its arguments. Matrix multiplication is an example. De ...
In many areas, such as
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, ...
, is reserved for a
sesquilinear 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 in each of its arguments, but a sesquilinear form allows o ...
defined on .
\langle \cdot, \cdot \rangle : V \times V^* \to F called the
natural pairing In mathematics, a dual system, dual pair, or duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces X and Y over \mathbb and a non-degenerate bilinear map b : X \times Y \to \mathbb. Duality theory, the study of dual ...
.


Finite-dimensional case

If ''V'' is finite-dimensional, then ''V''∗ has the same dimension as ''V''. Given 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 ...
in ''V'', it is possible to construct a specific basis in ''V''∗, 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 th ...
. This dual basis is a set of linear functionals on ''V'', defined by the relation : \mathbf^i(c^1 \mathbf_1+\cdots+c^n\mathbf_n) = c^i, \quad i=1,\ldots,n for any choice of coefficients . In particular, letting in turn each one of those coefficients be equal to one and the other coefficients zero, gives the system of equations : \mathbf^i(\mathbf_j) = \delta^_ where \delta^_ 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 &\ ...
symbol. This property is referred to as ''bi-orthogonality property''. Consider the basis of V. Let be defined as the following: \mathbf^i(c^1 \mathbf_1+\cdots+c^n\mathbf_n) = c^i, \quad i=1,\ldots,n . We have: # e^i , i=1, 2, \dots, n, are linear functionals. Indeed, for x,y \in V such as x= \alpha_1e_1 + \dots + \alpha_ne_n and y = \beta_1e_1 + \dots + \beta_n e_n (i.e, e^i(x)=\alpha_i and e^i(y)=\beta_i). Then, x+\lambda y=(\alpha_1+\lambda \beta_1)e_1 + \dots + (\alpha_n+\lambda\beta_n)e_n and e^i(x+\lambda y)=\alpha_i+\lambda\beta_i=e^i(x)+\lambda e^i(y) . Therefore, e^i \in V^* for i= 1, 2, \dots, n . # Suppose \lambda_1 e^1 + \cdots + \lambda_n e^n =0 \in V^*. Applying this functional on the basis vectors of V successively, lead us to \lambda_1=\lambda_2= \dots=\lambda_n=0 (The functional applied in e_i results in \lambda_i ). Therefore, is l.i. on V^* . #Lastly, consider g \in V^* . Then : g(x)=g(\alpha_1e_1 + \dots + \alpha_ne_n)=\alpha_1g(e_1) + \dots + \alpha_ng(e_n)=e^1(x)g(e_1) + \dots + e^n(x)g(e_n) and generates V^*. Hence, it is the basis of V^*. For example, if ''V'' is R2, let its basis be chosen as . The basis vectors are not orthogonal to each other. Then, e1 and e2 are
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 ...
s (functions that map a vector to a scalar) such that , , , and . (Note: The superscript here is the index, not an exponent.) This system of equations can be expressed using matrix notation as : \begin e_ & e_ \\ e_ & e_ \end \begin e^ & e^ \\ e^ & e^ \end = \begin 1 & 0 \\ 0 & 1 \end. Solving this equation shows the dual basis to be . Because e1 and e2 are functionals, they can be rewritten as e1(''x'', ''y'') = 2''x'' and e2(''x'', ''y'') = −''x'' + ''y''. In general, when ''V'' is R''n'', if E = (e1, ..., e''n'') is a matrix whose columns are the basis vectors and Ê = (e1, ..., e''n'') is a matrix whose columns are the dual basis vectors, then :E^T \hat = I_n, where ''I''''n'' is an identity matrix of order . The biorthogonality property of these two basis sets allows any point x ∈ ''V'' to be represented as :\mathbf = \sum_i \langle\mathbf,\mathbf^i \rangle \mathbf_i = \sum_i \langle \mathbf, \mathbf_i \rangle \mathbf^i, even when the basis vectors are not orthogonal to each other. Strictly speaking, the above statement only makes sense once the inner product \langle \cdot, \cdot \rangle and the corresponding duality pairing are introduced, as described below in '. In particular, R''n'' can be interpreted as the space of columns of
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, its dual space is typically written as the space of ''rows'' of real numbers. Such a row acts on R''n'' as a linear functional by ordinary
matrix multiplication 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 ...
. This is because a functional maps every -vector ''x'' into a real number ''y''. Then, seeing this functional as a matrix ''M'', and ''x'', ''y'' as a matrix and a matrix (trivially, a real number) respectively, if then, by dimension reasons, ''M'' must be a matrix; that is, ''M'' must be a row vector. If ''V'' consists of the space of geometrical
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
s in the plane, then the level curves of an element of ''V''∗ form a family of parallel lines in ''V'', because the range is 1-dimensional, so that every point in the range is a multiple of any one nonzero element. So an element of ''V''∗ can be intuitively thought of as a particular family of parallel lines covering the plane. To compute the value of a functional on a given vector, it suffices to determine which of the lines the vector lies on. Informally, this "counts" how many lines the vector crosses. More generally, if ''V'' is a vector space of any dimension, then the level sets of a linear functional in ''V''∗ are parallel hyperplanes in ''V'', and the action of a linear functional on a vector can be visualized in terms of these hyperplanes.


Infinite-dimensional case

If ''V'' is not finite-dimensional but has 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 ...
Several assertions in this article require the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
for their justification. The axiom of choice is needed to show that an arbitrary vector space has a basis: in particular it is needed to show that RN has a basis. It is also needed to show that the dual of an infinite-dimensional vector space ''V'' is nonzero, and hence that the natural map from ''V'' to its double dual is injective.
e''α'' indexed by an infinite set ''A'', then the same construction as in the finite-dimensional case yields
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
elements e''α'' () of the dual space, but they will not form a basis. For instance, the space R∞, whose elements are those
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s of real numbers that contain only finitely many non-zero entries, which has a basis indexed by the natural numbers N: for , e''i'' is the sequence consisting of all zeroes except in the ''i''-th position, which is ''1''. The dual space of R∞ is (isomorphic to) RN, the space of ''all'' sequences of real numbers: each real sequence (''an'') defines a function where the element (''xn'') of R∞ is sent to the number :\sum_n a_nx_n, which is a finite sum because there are only finitely many nonzero ''xn''. The
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of R∞ is countably infinite, whereas RN does not have a countable basis. This observation generalizes to any infinite-dimensional vector space ''V'' over any field ''F'': a choice of basis identifies ''V'' with the space (''FA'')0 of functions such that is nonzero for only finitely many , where such a function ''f'' is identified with the vector :\sum_ f_\alpha\mathbf_\alpha in ''V'' (the sum is finite by the assumption on ''f'', and any may be written in this way by the definition of the basis). The dual space of ''V'' may then be identified with the space ''FA'' of ''all'' functions from ''A'' to ''F'': a linear functional ''T'' on ''V'' is uniquely determined by the values it takes on the basis of ''V'', and any function (with ) defines a linear functional ''T'' on ''V'' by :T\left (\sum_ f_\alpha \mathbf_\alpha\right) = \sum_ f_\alpha T(e_\alpha) = \sum_ f_\alpha \theta_\alpha. Again the sum is finite because ''fα'' is nonzero for only finitely many ''α''. The set (''F''''A'')0 may be identified (essentially by definition) with the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of infinitely many copies of ''F'' (viewed as a 1-dimensional vector space over itself) indexed by ''A'', i.e. there are linear isomorphisms : V\cong (F^A)_0\cong\bigoplus_ F. On the other hand, ''FA'' is (again by definition), the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of infinitely many copies of ''F'' indexed by ''A'', and so the identification :V^* \cong \left (\bigoplus_F\right )^* \cong \prod_F^* \cong \prod_F \cong F^A is a special case of a general result relating direct sums (of modules) to direct products. Considering
cardinal numbers In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ...
, denoted here as
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
s, one has thus for a -vector space that has an infinite basis :, V, =\max(, F, , , A, ) < , V^\ast, =, F, ^. It follows that, if a vector space is not finite-dimensional, then the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
implies that the algebraic dual space is ''always'' of larger dimension (as a cardinal number) than the original vector space (since, if two bases have the same cardinality, the spanned vector spaces have the same cardinality). This is in contrast to the case of the continuous dual space, discussed below, which may be
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 the original vector space even if the latter is infinite-dimensional.


Bilinear products and dual spaces

If ''V'' is finite-dimensional, then ''V'' is isomorphic to ''V''∗. But there is in general no
natural isomorphism In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural ...
between these two spaces. Any
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 ''V'' gives a mapping of ''V'' into its dual space via :v\mapsto \langle v, \cdot\rangle where the right hand side is defined as the functional on ''V'' taking each to . In other words, the bilinear form determines a linear mapping :\Phi_ : V\to V^* defined by :\left Phi_(v), w\right= \langle v, w\rangle. If the bilinear form is
nondegenerate In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case. T ...
, then this is an isomorphism onto a subspace of ''V''∗. If ''V'' is finite-dimensional, then this is an isomorphism onto all of ''V''∗. Conversely, any isomorphism \Phi from ''V'' to a subspace of ''V''∗ (resp., all of ''V''∗ if ''V'' is finite dimensional) defines a unique nondegenerate bilinear form on ''V'' by : \langle v, w \rangle_\Phi = (\Phi (v))(w) =
Phi (v), w Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
\, Thus there is a one-to-one correspondence between isomorphisms of ''V'' to a subspace of (resp., all of) ''V''∗ and nondegenerate bilinear forms on ''V''. If the vector space ''V'' is over the
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
field, then sometimes it is more natural to consider
sesquilinear 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 in each of its arguments, but a sesquilinear form allows o ...
s instead of bilinear forms. In that case, a given sesquilinear form determines an isomorphism of ''V'' with the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of the dual space : \Phi_ : V\to \overline. The conjugate of the dual space \overline can be identified with the set of all additive complex-valued functionals such that : f(\alpha v) = \overlinef(v).


Injection into the double-dual

There is a
natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are p ...
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
\Psi from V into the double dual V^=\, defined by (\Psi(v))(\varphi)=\varphi(v) for all v\in V, \varphi\in V^*. In other words, if \mathrm_v:V^*\to F is the evaluation map defined by \varphi \mapsto \varphi(v), then \Psi: V \to V^ is defined as the map v\mapsto\mathrm_v. This map \Psi is always
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
; it is 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 ...
if and only if V is finite-dimensional. Indeed, the isomorphism of a finite-dimensional vector space with its double dual is an archetypal example of a
natural isomorphism In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural ...
. Infinite-dimensional Hilbert spaces are not isomorphic to their algebraic double duals, but instead to their continuous double duals.


Transpose of a linear map

If 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 ...
, then the ''
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
'' (or ''dual'') is defined by : f^*(\varphi) = \varphi \circ f \, for every ''\varphi \in W^*''. The resulting functional ''f^* (\varphi)'' in ''V^*'' is called the ''
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
'' of ''\varphi'' along ''f''. The following identity holds for all ''\varphi \in W^*'' and ''v \in V'': : ^*(\varphi),\, v= varphi,\, f(v) where the bracket ·,·on the left is the natural pairing of ''V'' with its dual space, and that on the right is the natural pairing of ''W'' with its dual. This identity characterizes the transpose, and is formally similar to the definition of the
adjoint In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...
. The assignment produces an
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
linear map between the space of linear operators from ''V'' to ''W'' and the space of linear operators from ''W'' to ''V''; this homomorphism is 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 ...
if and only if ''W'' is finite-dimensional. If then the space of linear maps is actually an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
under
composition of maps In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
, and the assignment is then an
antihomomorphism In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is a bijective antihomomorphism, i.e. an antiisomorphism, from a set to itself. From ...
of algebras, meaning that . In the language of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, taking the dual of vector spaces and the transpose of linear maps is therefore a
contravariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
from the category of vector spaces over ''F'' to itself. It is possible to identify (''f'') with ''f'' using the natural injection into the double dual. If the linear map ''f'' is represented by the
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
''A'' with respect to two bases of ''V'' and ''W'', then ''f'' is represented by the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
matrix ''A''T with respect to the dual bases of ''W'' and ''V'', hence the name. Alternatively, as ''f'' is represented by ''A'' acting on the left on column vectors, ''f'' is represented by the same matrix acting on the right on row vectors. These points of view are related by the canonical inner product on R''n'', which identifies the space of column vectors with the dual space of row vectors.


Quotient spaces and annihilators

Let ''S'' be a subset of ''V''. The annihilator of ''S'' in ''V''∗, denoted here ''S'', is the collection of linear functionals such that for all . That is, ''S'' consists of all linear functionals such that the restriction to ''S'' vanishes: . Within finite dimensional vector spaces, the annihilator is dual to (isomorphic to) the
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
. The annihilator of a subset is itself a vector space. The annihilator of the zero vector is the whole dual space: \^0 = V^*, and the annihilator of the whole space is just the zero covector: V^0 = \ \subseteq V^*. Furthermore, the assignment of an annihilator to a subset of ''V'' reverses inclusions, so that if , then : 0 \subseteq T^0 \subseteq S^0 \subseteq V^* . If ''A'' and ''B'' are two subsets of ''V'' then : A^0 + B^0 \subseteq (A \cap B)^0, and equality holds provided ''V'' is finite-dimensional. If ''Ai'' is any family of subsets of ''V'' indexed by ''i'' belonging to some index set ''I'', then : \left( \bigcup_ A_i \right)^0 = \bigcap_ A_i^0 . In particular if ''A'' and ''B'' are subspaces of ''V'' then : (A + B)^0 = A^0 \cap B^0 . If ''V'' is finite-dimensional and ''W'' is 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, li ...
, then : W^ = W after identifying ''W'' with its image in the second dual space under the double duality isomorphism . In particular, forming the annihilator is a
Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the funda ...
on the lattice of subsets of a finite-dimensional vector space. If ''W'' is a subspace of ''V'' then the quotient space ''V''/''W'' is a vector space in its own right, and so has a dual. By the
first isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist f ...
, a functional factors through ''V''/''W'' if and only if ''W'' is in the
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 ...
of ''f''. There is thus an isomorphism : (V/W)^* \cong W^0 . As a particular consequence, if ''V'' is a
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of two subspaces ''A'' and ''B'', then ''V''∗ is a direct sum of ''A'' and ''B''.


Dimensional analysis

The dual space is analogous to a "negative"-dimensional space. Most simply, since a vector v \in V can be paired with a covector \varphi \in V^* by the natural pairing \langle x, \varphi \rangle := \varphi (x) \in F to obtain a scalar, a covector can "cancel" the dimension of a vector, similar to
reducing a fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
. Thus while the direct sum V \oplus V^* is an -dimensional space (if is -dimensional), behaves as an -dimensional space, in the sense that its dimensions can be canceled against the dimensions of . This is formalized by
tensor contraction In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tens ...
. This arises in physics via
dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as m ...
, where the dual space has inverse units. Under the natural pairing, these units cancel, and the resulting scalar value is
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
, as expected. For example in (continuous)
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...
, or more broadly
time–frequency analysis In signal processing, time–frequency analysis comprises those techniques that study a signal in both the time and frequency domains ''simultaneously,'' using various time–frequency representations. Rather than viewing a 1-dimensional signal (a ...
:To be precise, continuous Fourier analysis studies the space of functionals with domain a vector space and the space of functionals on the dual vector space. given a one-dimensional vector space with a
unit of time A unit of time is any particular time interval, used as a standard way of measuring or expressing duration. The base unit of time in the International System of Units (SI) and by extension most of the Western world, is the second, defined as a ...
, the dual space has units of
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
: occurrences ''per'' unit of time (units of ). For example, if time is measured in
second The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ...
s, the corresponding dual unit is the
inverse second The inverse second or reciprocal second (s−1) is a unit of frequency, defined as the multiplicative inverse of the second (a unit of time). It is dimensionally equivalent to: * the unit hertz – the SI unit for frequency * the unit radian per ...
: over the course of 3 seconds, an event that occurs 2 times per second occurs a total of 6 times, corresponding to 3s \cdot 2s^ = 6. Similarly, if the primal space measures length, the dual space measures
inverse length Reciprocal length or inverse length is a quantity or measurement used in several branches of science and mathematics. As the reciprocal of length, common units used for this measurement include the reciprocal metre or inverse metre (symbol: m&min ...
.


Continuous dual space

When dealing with
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, the
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 from the space into the base field \mathbb = \Complex (or \R) are particularly important. This gives rise to the notion of the "continuous dual space" or "topological dual" which is a linear subspace of the algebraic dual space V^*, denoted by V'. For any ''finite-dimensional'' normed vector space or topological vector space, such as Euclidean ''n-''space, the continuous dual and the algebraic dual coincide. This is however false for any infinite-dimensional normed space, as shown by the example of
discontinuous linear map In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see linear approximation). If the spaces involved ar ...
s. Nevertheless, in the theory of
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 the terms "continuous dual space" and "topological dual space" are often replaced by "dual space". For 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 ...
V its ''continuous dual space'', or ''topological dual space'', or just ''dual space'' (in the sense of the theory of topological vector spaces) V' is defined as the space of all continuous linear functionals \varphi:V\to. Important examples for continuous dual spaces are the space of compactly supported
test functions 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 ...
\mathcal and its dual \mathcal', the space of arbitrary distributions (generalized functions); the space of arbitrary test functions \mathcal and its dual \mathcal', the space of compactly supported distributions; and the space of rapidly decreasing test functions \mathcal, the
Schwartz space In mathematics, Schwartz space \mathcal is the function space of all Function (mathematics), functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. T ...
, and its dual \mathcal', the space of
tempered distributions 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 derivative, differentiate functions whose de ...
(slowly growing distributions) in the theory of
generalized functions 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 ...
.


Properties

If is a Hausdorff
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), then the continuous dual space of is identical to the continuous dual space of the completion of .


Topologies on the dual

There is a standard construction for introducing a topology on the continuous dual V' of a topological vector space V. Fix a collection \mathcal of bounded subsets of V. This gives the topology on V of uniform convergence on sets from \mathcal, or what is the same thing, the topology generated by
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
s of the form :\, \varphi\, _A = \sup_ , \varphi(x), , where \varphi is a continuous linear functional on V, and A runs over the class \mathcal. This means that a net of functionals \varphi_i tends to a functional \varphi in V' if and only if :\text A\in\mathcal\qquad \, \varphi_i-\varphi\, _A = \sup_ , \varphi_i(x)-\varphi(x), \underset 0. Usually (but not necessarily) the class \mathcal is supposed to satisfy the following conditions: * Each point x of V belongs to some set A\in\mathcal: ::\text x \in V\quad \text A \in \mathcal\quad \text x \in A. * Each two sets A \in \mathcal and B \in \mathcal are contained in some set C \in \mathcal: ::\text A, B \in \mathcal\quad \text C \in \mathcal\quad \text A \cup B \subseteq C. * \mathcal is closed under the operation of multiplication by scalars: ::\text A \in \mathcal\quad \text \lambda \in \quad \text \lambda \cdot A \in \mathcal. If these requirements are fulfilled then the corresponding topology on V' is Hausdorff and the sets :U_A ~=~ \left \,\qquad \text A \in \mathcal form its local base. Here are the three most important special cases. * The
strong topology In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to: * the final topology on the disjoint union * the to ...
on V' is the topology of uniform convergence on bounded subsets in V (so here \mathcal can be chosen as the class of all bounded subsets in V). If V is a
normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
(for example, 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 ...
or a
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 ...
) then the strong topology on V' is normed (in fact a Banach space if the field of scalars is complete), with the norm ::\, \varphi\, = \sup_ , \varphi(x), . * The stereotype topology on V' is the topology of uniform convergence on totally bounded sets in V (so here \mathcal can be chosen as the class of all totally bounded subsets in V). * The
weak topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
on V' is the topology of uniform convergence on finite subsets in V (so here \mathcal can be chosen as the class of all finite subsets in V). Each of these three choices of topology on V' leads to a variant of
reflexivity property In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal to ...
for topological vector spaces: * If V' is endowed with the
strong topology In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to: * the final topology on the disjoint union * the to ...
, then the corresponding notion of reflexivity is the standard one: the spaces reflexive in this sense are just called ''reflexive''. * If V' is endowed with the stereotype dual topology, then the corresponding reflexivity is presented in the theory of stereotype spaces: the spaces reflexive in this sense are called ''stereotype''. * If V' is endowed with the
weak topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
, then the corresponding reflexivity is presented in the theory of
dual pair In mathematics, a dual system, dual pair, or duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces X and Y over \mathbb and a non-degenerate bilinear map b : X \times Y \to \mathbb. Duality theory, the study of dual ...
s: the spaces reflexive in this sense are arbitrary (Hausdorff) locally convex spaces with the weak topology.


Examples

Let 1 < ''p'' < ∞ be a real number and consider the Banach space '' â„“ p'' of all
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s for which :\, \mathbf\, _p = \left ( \sum_^\infty , a_n, ^p \right) ^ < \infty. Define the number ''q'' by . Then the continuous dual of ''â„“'' ''p'' is naturally identified with ''â„“'' ''q'': given an element \varphi \in (\ell^p)', the corresponding element of is the sequence (\varphi(\mathbf _n)) where \mathbf _n denotes the sequence whose -th term is 1 and all others are zero. Conversely, given an element , the corresponding continuous linear functional ''\varphi'' on is defined by :\varphi (\mathbf) = \sum_n a_n b_n for all (see Hölder's inequality). In a similar manner, the continuous dual of is naturally identified with (the space of bounded sequences). Furthermore, the continuous duals of the Banach spaces ''c'' (consisting of all convergent sequences, with the
supremum norm In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when the ...
) and ''c''0 (the sequences converging to zero) are both naturally identified with . 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 ...
, the continuous dual of a Hilbert space is again a Hilbert space which is anti-isomorphic to the original space. This gives rise to the
bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathema ...
used by physicists in the mathematical formulation of
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, ...
. By the
Riesz–Markov–Kakutani representation theorem In mathematics, the Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory. The theorem is named for who introduced it for continuo ...
, the continuous dual of certain spaces of continuous functions can be described using measures.


Transpose of a continuous linear map

If is a continuous linear map between two topological vector spaces, then the (continuous) transpose is defined by the same formula as before: :T'(\varphi) = \varphi \circ T, \quad \varphi \in W'. The resulting functional is in . The assignment produces a linear map between the space of continuous linear maps from ''V'' to ''W'' and the space of linear maps from to . When ''T'' and ''U'' are composable continuous linear maps, then :(U \circ T)' = T' \circ U'. When ''V'' and ''W'' are normed spaces, the norm of the transpose in is equal to that of ''T'' in . Several properties of transposition depend upon the
Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
. For example, the bounded linear map ''T'' has dense range if and only if the transpose is injective. When ''T'' is a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
linear map between two Banach spaces ''V'' and ''W'', then the transpose is compact. This can be proved using the
Arzelà–Ascoli theorem The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded inter ...
. When ''V'' is a Hilbert space, there is an antilinear isomorphism ''iV'' from ''V'' onto its continuous dual . For every bounded linear map ''T'' on ''V'', the transpose and the
adjoint In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...
operators are linked by :i_V \circ T^* = T' \circ i_V. When ''T'' is a continuous linear map between two topological vector spaces ''V'' and ''W'', then the transpose is continuous when and are equipped with "compatible" topologies: for example, when for and , both duals have the
strong topology In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to: * the final topology on the disjoint union * the to ...
of uniform convergence on bounded sets of ''X'', or both have the weak-∗ topology of pointwise convergence on ''X''. The transpose is continuous from to , or from to .


Annihilators

Assume that ''W'' is a closed linear subspace of a normed space ''V'', and consider the annihilator of ''W'' in , :W^\perp = \. Then, the dual of the quotient can be identified with ''W''⊥, and the dual of ''W'' can be identified with the quotient . Indeed, let ''P'' denote the canonical
surjection 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 ...
from ''V'' onto the quotient ; then, the transpose is an isometric isomorphism from into , with range equal to ''W''⊥. If ''j'' denotes the injection map from ''W'' into ''V'', then the kernel of the transpose is the annihilator of ''W'': :\ker (j') = W^\perp and it follows from the
Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
that induces an isometric isomorphism .


Further properties

If the dual of a normed space is separable, then so is the space itself. The converse is not true: for example, the space is separable, but its dual is not.


Double dual

In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator from a normed space ''V'' into its continuous double dual , defined by : \Psi(x)(\varphi) = \varphi(x), \quad x \in V, \ \varphi \in V' . As a consequence of the
Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
, this map is in fact an
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
, meaning for all . Normed spaces for which the map Ψ is a
bijection 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 s ...
are called reflexive. When ''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 ...
then Ψ(''x'') can still be defined by the same formula, for every , however several difficulties arise. First, when ''V'' is not locally convex, the continuous dual may be equal to and the map Ψ trivial. However, if ''V'' is Hausdorff and locally convex, the map Ψ is injective from ''V'' to the algebraic dual of the continuous dual, again as a consequence of the Hahn–Banach theorem.If ''V'' is locally convex but not Hausdorff, the
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 ...
of Ψ is the smallest closed subspace containing .
Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual , so that the continuous double dual is not uniquely defined as a set. Saying that Ψ maps from ''V'' to , or in other words, that Ψ(''x'') is continuous on for every , is a reasonable minimal requirement on the topology of , namely that the evaluation mappings : \varphi \in V' \mapsto \varphi(x), \quad x \in V , be continuous for the chosen topology on . Further, there is still a choice of a topology on , and continuity of Ψ depends upon this choice. As a consequence, defining reflexivity in this framework is more involved than in the normed case.


See also

*
Covariance and contravariance of vectors In physics, especially in 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. In modern mathematical notation ...
*
Dual module In mathematics, the dual module of a left (respectively right) module ''M'' over a ring ''R'' is the set of module homomorphisms from ''M'' to ''R'' with the pointwise right (respectively left) module structure. The dual module is typically denote ...
*
Dual norm In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space. Definition Let X be a normed vector space with norm \, \cdot\, and let X^* denote its continuous dual space. The dual n ...
*
Duality (mathematics) In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the ...
*
Duality (projective geometry) In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of dua ...
*
Pontryagin duality In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), ...
*
Reciprocal lattice In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...
– dual space basis, in crystallography


Notes


References


Bibliography

* * * * * * * * . * * * * * * * *


External links

* Functional analysis {{DEFAULTSORT:Dual Space Linear algebra
Space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...