The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called thealgebraic dual space.When defined for atopological vector space, there is a subspace of the dual space, corresponding tocontinuous linear functionals, called thecontinuous dual space.
Dual vector spaces find application in many branches of mathematics that use vector spaces, such as intensor analysis withfinite-dimensional vector spaces.When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describemeasures,distributions, andHilbert spaces. Consequently, the dual space is an important concept infunctional analysis.
Early terms fordual includepolarer Raum [Hahn 1927],espace conjugué,adjoint space [Alaoglu 1940], andtransponierter Raum [Schauder 1930] and [Banach 1932]. The termdual is due toBourbaki 1938.[1]
Given anyvector space over afield, the(algebraic) dual space[2] (alternatively denoted by[3] or[4][5])[nb 1] is defined as the set of alllinear maps (linear functionals). Since linear maps are vector spacehomomorphisms, the dual space may be denoted.[3]The dual space itself becomes a vector space over when equipped with an addition and scalar multiplication satisfying:for all,, and. For example, if we express the vector space as the set of vectors (where and are real numbers), the function
is an element of, since it is-linear and maps vectors in to elements of.
Elements of the algebraic dual space are sometimes calledcovectors,one-forms, orlinear forms.
The pairing of a functional in the dual space and an element of is sometimes denoted by a bracket:[6]or.[7] This pairing defines a nondegeneratebilinear mapping[nb 2] called thenatural pairing.
Given a vector space and a basis on that space, one can define alinearly independent set in called thedual set. Each vector in corresponds to a unique vector in the dual set. This correspondence yields an injection.
If is finite-dimensional, the dual set is a basis, called thedual basis, and the injection is anisomorphism.
If is finite-dimensional and has a basis, in, the dual basis is a set of linear functionals on, defined by the relationfor 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 equationswhere is theKronecker delta symbol. This property is referred to as thebi-orthogonality property.
Proof
Consider the basis of V. Let be defined as the following:
.
These are a basis of because:
The are linear functionals, which map such as and to scalars and. Then also, and. Therefore, for.
Suppose. Applying this functional on the basis vectors of successively, lead us to (The functional applied in results in). Therefore, is linearly independent on.
Lastly, consider. Then so. So generates.
Hence, it is a basis of.
For example, if is, let its basis be chosen as. The basis vectors are not orthogonal to each other. Then, and areone-forms (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 asSolving for the unknown values in the first matrix shows the dual basis to be. Because and are functionals, they can be rewritten as and.
In general, when is, if is a matrix whose columns are the basis vectors and is a matrix whose columns are the dual basis vectors, thenwhere is theidentity matrix of order. The biorthogonality property of these two basis sets allows any point to be represented as
even when the basis vectors are not orthogonal to each other. Strictly speaking, the above statement only makes sense once the inner product and the corresponding duality pairing are introduced, as described below in§ Bilinear products and dual spaces.
In particular, can be interpreted as the space of columns ofreal numbers, its dual space is typically written as the space ofrows of real numbers. Such a row acts on as a linear functional by ordinarymatrix multiplication. This is because a functional maps every-vector into a real number. Then, seeing this functional as a matrix, and as an matrix, and a matrix (trivially, a real number) respectively, if then, by dimension reasons, must be a matrix; that is, must be a row vector.
If consists of the space of geometricalvectors in the plane, then thelevel curves of an element of form a family of parallel lines in, 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 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 is a vector space of any dimension, then thelevel sets of a linear functional in are parallel hyperplanes in, and the action of a linear functional on a vector can be visualized in terms of these hyperplanes.[8]
If is not finite-dimensional but has abasis[nb 3] indexed by an infinite set, then the same construction as in the finite-dimensional case yieldslinearly independent elements () of the dual space, but they will not form a basis.
For instance, consider the space, whose elements are thosesequences of real numbers that contain only finitely many non-zero entries, which has a basis indexed by the natural numbers. For, is the sequence consisting of all zeroes except in the-th position, which is 1.The dual space of is (isomorphic to), the space ofall sequences of real numbers: each real sequence defines a function where the element of is sent to the number
which is a finite sum because there are only finitely many nonzero. Thedimension of iscountably infinite, whereas does not have a countable basis.
This observation generalizes to any[nb 3] infinite-dimensional vector space over any field: a choice of basis identifies with the space of functions such that is nonzero for only finitely many, where such a function is identified with the vector
in (the sum is finite by the assumption on, and any may be written uniquely in this way by the definition of the basis).
The dual space of may then be identified with the space ofall functions from to: a linear functional on is uniquely determined by the values it takes on the basis of, and any function (with) defines a linear functional on by
Again, the sum is finite because is nonzero for only finitely many.
The set may be identified (essentially by definition) with thedirect sum of infinitely many copies of (viewed as a 1-dimensional vector space over itself) indexed by, i.e. there are linear isomorphisms
On the other hand, is (again by definition), thedirect product of infinitely many copies of indexed by, and so the identificationis a special case of ageneral result relating direct sums (ofmodules) to direct products.
If a vector space is not finite-dimensional, then its (algebraic) dual space isalways of larger dimension (as acardinal number) than the original vector space. This is in contrast to the case of the continuous dual space, discussed below, which may beisomorphic to the original vector space even if the latter is infinite-dimensional.
The proof of this inequality between dimensions results from the following.
If is an infinite-dimensional-vector space, the arithmetical properties ofcardinal numbers implies thatwhere cardinalities are denoted asabsolute values. For proving that it suffices to prove that which can be done with an argument similar toCantor's diagonal argument.[9] The exact dimension of the dual is given by theErdős–Kaplansky theorem.
If is finite-dimensional, then is isomorphic to. But there is in general nonatural isomorphism between these two spaces.[10] Anybilinear form on gives a mapping of into its dual space via
where the right hand side is defined as the functional on taking each to. In other words, the bilinear form determines a linear mapping
defined by
If the bilinear form isnondegenerate, then this is an isomorphism onto a subspace of.If is finite-dimensional, then this is an isomorphism onto all of. Conversely, any isomorphism from to a subspace of (resp., all of if is finite dimensional) defines a unique nondegenerate bilinear form on by
Thus there is a one-to-one correspondence between isomorphisms of to a subspace of (resp., all of) and nondegenerate bilinear forms on.
If the vector space is over thecomplex field, then sometimes it is more natural to considersesquilinear forms instead of bilinear forms.In that case, a given sesquilinear form determines an isomorphism of with thecomplex conjugate of the dual space
The conjugate of the dual space can be identified with the set of all additive complex-valued functionals such that
There is anaturalhomomorphism from into the double dual, defined by for all. In other words, if is the evaluation map defined by, then is defined as the map. This map is alwaysinjective;[nb 3] and it is always anisomorphism if is finite-dimensional.[11]Indeed, the isomorphism of a finite-dimensional vector space with its double dual is an archetypal example of anatural isomorphism.Infinite-dimensional Hilbert spaces are not isomorphic to their algebraic double duals, but instead to their continuous double duals.
If is alinear map, then thetranspose (ordual) is defined byfor every. The resulting functional in is called thepullback of along.
The following identity holds for all and:where the bracket on the left is the natural pairing of with its dual space, and that on the right is the natural pairing of with its dual. This identity characterizes the transpose,[12] and is formally similar to the definition of theadjoint.
The assignment produces aninjective linear map between the space of linear operators from to and the space of linear operators from to; this homomorphism is anisomorphism if and only if is finite-dimensional.If then the space of linear maps is actually analgebra undercomposition of maps, and the assignment is then anantihomomorphism of algebras, meaning that.In the language ofcategory theory, taking the dual of vector spaces and the transpose of linear maps is therefore acontravariant functor from the category of vector spaces over to itself.It is possible to identify with using the natural injection into the double dual.
If the linear map is represented by thematrix with respect to two bases of and, then is represented by thetranspose matrix with respect to the dual bases of and, hence the name.Alternatively, as is represented by acting on the left on column vectors, 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, which identifies the space of column vectors with the dual space of row vectors.
Let be a subset of.Theannihilator of in, denoted here, is the collection of linear functionals such that for all.That is, consists of all linear functionals such that the restriction to vanishes:.Within finite dimensional vector spaces, the annihilator is dual to (isomorphic to) theorthogonal complement.
The annihilator of a subset is itself a vector space.The annihilator of the zero vector is the whole dual space:, and the annihilator of the whole space is just the zero covector:.Furthermore, the assignment of an annihilator to a subset of reverses inclusions, so that if, then
If and are two subsets of thenIf is any family of subsets of indexed by belonging to some index set, thenIn particular if and are subspaces of thenand[nb 3]
If is finite-dimensional and is avector subspace, thenafter identifying with its image in the second dual space under the double duality isomorphism. In particular, forming the annihilator is aGalois connection on the lattice of subsets of a finite-dimensional vector space.
If is a subspace of then thequotient space is a vector space in its own right, and so has a dual. By thefirst isomorphism theorem, a functional factors through if and only if is in thekernel of. There is thus an isomorphism
As a particular consequence, if is adirect sum of two subspaces and, then is a direct sum of and.
The dual space is analogous to a "negative"-dimensional space. Most simply, since a vector can be paired with a covector by the natural pairing to obtain a scalar, a covector can "cancel" the dimension of a vector, similar toreducing a fraction. Thus while the direct sum is a-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 bytensor contraction.
This arises in physics viadimensional analysis, where the dual space has inverse units.[13] Under the natural pairing, these units cancel, and the resulting scalar value isdimensionless, as expected. For example, in (continuous)Fourier analysis, or more broadlytime–frequency analysis:[nb 4] given a one-dimensional vector space with aunit of time, the dual space has units offrequency: occurrencesper unit of time (units of). For example, if time is measured inseconds, the corresponding dual unit is theinverse second: over the course of 3 seconds, an event that occurs 2 times per second occurs a total of 6 times, corresponding to. Similarly, if the primal space measures length, the dual space measuresinverse length.
When dealing withtopological vector spaces, thecontinuous linear functionals from the space into the base field (or) 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, denoted by.For anyfinite-dimensional normed vector space or topological vector space, such asEuclideann-space, the continuous dual and the algebraic dual coincide.This is however false for any infinite-dimensional normed space, as shown by the example ofdiscontinuous linear maps.Nevertheless, in the theory oftopological vector spaces the terms "continuous dual space" and "topological dual space" are often replaced by "dual space".
For atopological vector space itscontinuous dual space,[14] ortopological dual space,[15] or justdual space[14][15][16][17] (in the sense of the theory of topological vector spaces) is defined as the space of all continuous linear functionals.
Important examples for continuous dual spaces are the space of compactly supportedtest functions and its dual the space of arbitrarydistributions (generalized functions); the space of arbitrary test functions and its dual the space of compactly supported distributions; and the space of rapidly decreasing test functions theSchwartz space, and its dual the space oftempered distributions (slowly growing distributions) in the theory ofgeneralized functions.
There is a standard construction for introducing a topology on the continuous dual of a topological vector space. Fix a collection ofbounded subsets of.This gives the topology on ofuniform convergence on sets from or what is the same thing, the topology generated byseminorms of the form
where is a continuous linear functional on, and runs over the class
This means that a net of functionals tends to a functional in if and only if
Usually (but not necessarily) the class is supposed to satisfy the following conditions:
Each point of belongs to some set.
Each two sets and are contained in some set.
is closed under the operation of multiplication by scalars.
If these requirements are fulfilled then the corresponding topology on is Hausdorff and the sets
form its local base.
Here are the three most important special cases.
Thestrong topology on is the topology of uniform convergence onbounded subsets in (so here can be chosen as the class of all bounded subsets in).
If is anormed vector space (for example, aBanach space or aHilbert space) then the strong topology on is normed (in fact a Banach space if the field of scalars is complete), with the norm
Thestereotype topology on is the topology of uniform convergence ontotally bounded sets in (so here can be chosen as the class of all totally bounded subsets in).
Theweak topology on is the topology of uniform convergence on finite subsets in (so here can be chosen as the class of all finite subsets in).
Each of these three choices of topology on leads to a variant ofreflexivity property for topological vector spaces:
If is endowed with thestrong topology, then the corresponding notion of reflexivity is the standard one: the spaces reflexive in this sense are just calledreflexive.[18]
If is endowed with the stereotype dual topology, then the corresponding reflexivity is presented in the theory ofstereotype spaces: the spaces reflexive in this sense are calledstereotype.
If is endowed with theweak topology, then the corresponding reflexivity is presented in the theory ofdual pairs:[19] the spaces reflexive in this sense are arbitrary (Hausdorff) locally convex spaces with the weak topology.[20]
Let 1 < < ∞ be a real number and consider the Banach spaceℓ p of allsequences for which
Define the number by. Then the continuous dual of is naturally identified with: given an element, the corresponding element of is the sequence where denotes the sequence whose-th term is 1 and all others are zero. Conversely, given an element, the corresponding continuous linear functional on is defined by
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 (consisting of allconvergent sequences, with thesupremum norm) and (the sequences converging to zero) are both naturally identified with.
If is a continuous linear map between two topological vector spaces, then the (continuous) transpose is defined by the same formula as before:
The resulting functional is in. The assignment produces a linear map between the space of continuous linear maps from to and the space of linear maps from to.When and are composable continuous linear maps, then
When and are normed spaces, the norm of the transpose in is equal to that of in.Several properties of transposition depend upon theHahn–Banach theorem.For example, the bounded linear map has dense rangeif and only if the transpose is injective.
When is acompact linear map between two Banach spaces and, then the transpose is compact.This can be proved using theArzelà–Ascoli theorem.
When is a Hilbert space, there is an antilinear isomorphism from onto its continuous dual.For every bounded linear map on, the transpose and theadjoint operators are linked by
When is a continuous linear map between two topological vector spaces and, then the transpose is continuous when and are equipped with "compatible" topologies: for example, when for and, both duals have thestrong topology of uniform convergence on bounded sets of, or both have the weak-∗ topology of pointwise convergence on.The transpose is continuous from to, or from to.
Assume that is a closed linear subspace of a normed space, and consider the annihilator of in,
Then, the dual of the quotient can be identified with, and the dual of can be identified with the quotient.[21]Indeed, let denote the canonicalsurjection from onto the quotient. Then the transpose is an isometric isomorphism from into, with range equal to.If denotes the injection map from into, then the kernel of the transpose is the annihilator of:and it follows from theHahn–Banach theorem that induces an isometric isomorphism.
If the dual of a normed space isseparable, then so is the space itself.The converse is not true: for example, the space is separable, but its dual is not.
This is anatural transformation of vector addition from a vector space to its double dual. denotes theordered pair of two vectors. The addition + sends and to. The addition induced by the transformation can be defined as for any in the dual space.
In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator from a normed space into its continuous double dual, defined by
When is atopological vector space then() can still be defined by the same formula, for every, however several difficulties arise.First, when is notlocally convex, the continuous dual may be equal to { 0 } and the map trivial.However, if isHausdorff and locally convex, the map is injective from to the algebraic dual of the continuous dual, again as a consequence of the Hahn–Banach theorem.[nb 5]
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 maps from to, or in other words, that is continuous on for every, is a reasonable minimal requirement on the topology of, namely that the evaluation mappings
be continuous for the chosen topology on. Further, there is still a choice of a topology on, and continuity of depends upon this choice.As a consequence, definingreflexivity in this framework is more involved than in the normed case.
^For used in this way, seeAn Introduction to Manifolds (Tu 2011, p. 19).This notation is sometimes used when is reserved for some other meaning.For instance, in the above text, is frequently used to denote the codifferential of, so that represents the pullback of the form.Halmos (1974, p. 20) uses to denote the algebraic dual of. However, other authors use for the continuous dual, while reserving for the algebraic dual (Trèves 2006, p. 35).
^abcdSeveral assertions in this article require theaxiom of choice 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 has a basis.It is also needed to show that the dual of an infinite-dimensional vector space is nonzero, and hence that the natural map from to its double dual is injective.
^To be precise, continuous Fourier analysis studies the space offunctionals with domain a vector space and the space of functionals on the dual vector space.
^If is locally convex but not Hausdorff, thekernel of is the smallest closed subspace containing {0}.
