Dual space
Inmathematics,anyvector spacehas a correspondingdual vector space(or justdual spacefor short) consisting of alllinear formsontogether with the vector space structure ofpointwiseaddition 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 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 intensoranalysis withfinite-dimensionalvector 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 fordualincludepolarer Raum[Hahn 1927],espace conjugué,adjoint space[Alaoglu 1940], andtransponierter Raum[Schauder 1930] and [Banach 1932]. The termdualis due toBourbaki1938.[1]
Algebraic dual space
[edit]Given anyvector spaceover 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 spaceitself becomes a vector space overwhen equipped with an addition and scalar multiplication satisfying:
for all,,and.
Elements of the algebraic dual spaceare sometimes calledcovectors,one-forms,orlinear forms.
The pairing of a functionalin the dual spaceand an elementofis sometimes denoted by a bracket:[6] or.[7]This pairing defines a nondegeneratebilinear mapping[nb 2]called thenatural pairing.
Finite-dimensional case
[edit]Ifis finite-dimensional, thenhas the same dimension as.Given abasisin,it is possible to construct a specific basis in,called thedual basis.This dual basis is a setof linear functionals on,defined by the relation
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
whereis theKronecker deltasymbol. This property is referred to as thebi-orthogonality property.
Proof
|
---|
Considerthe basis of V. Letbe defined as the following: . These are a basis ofbecause:
andgenerates.Hence, it is a basis of. |
For example, ifis,let its basis be chosen as.The basis vectors are not orthogonal to each other. Then,andareone-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 as
Solving for the unknown values in the first matrix shows the dual basis to be.Becauseandare functionals, they can be rewritten asand.
In general, whenis,ifis a matrix whose columns are the basis vectors andis a matrix whose columns are the dual basis vectors, then
whereis theidentity matrixof order.The biorthogonality property of these two basis sets allows any pointto 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 productand 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 ofrowsofreal numbers. Such a row acts onas a linear functional by ordinarymatrix multiplication.This is because a functional maps every-vectorinto a real number.Then, seeing this functional as a matrix,andas anmatrix, andamatrix (trivially, a real number) respectively, ifthen, by dimension reasons,must be amatrix; that is,must be a row vector.
Ifconsists of the space of geometricalvectorsin the plane, then thelevel curvesof an element ofform 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 ofcan 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, ifis a vector space of any dimension, then thelevel setsof a linear functional inare parallel hyperplanes in,and the action of a linear functional on a vector can be visualized in terms of these hyperplanes.[8]
Infinite-dimensional case
[edit]Ifis not finite-dimensional but has abasis[nb 3]indexed by an infinite set,then the same construction as in the finite-dimensional case yieldslinearly independentelements() of the dual space, but they will not form a basis.
For instance, consider the space,whose elements are thosesequencesof 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 ofis (isomorphic to),the space ofallsequences of real numbers: each real sequencedefines a function where the elementofis sent to the number
which is a finite sum because there are only finitely many nonzero.Thedimensionofiscountably infinite,whereasdoes not have a countable basis.
This observation generalizes to any[nb 3]infinite-dimensional vector spaceover any field:a choice of basisidentifieswith the spaceof functionssuch thatis nonzero for only finitely many,where such a functionis identified with the vector
in(the sum is finite by the assumption on,and anymay be written uniquely in this way by the definition of the basis).
The dual space ofmay then be identified with the spaceofallfunctions fromto:a linear functionalonis uniquely determined by the valuesit takes on the basis of,and any function(with) defines a linear functionalonby
Again, the sum is finite becauseis nonzero for only finitely many.
The setmay be identified (essentially by definition) with thedirect sumof 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 productof infinitely many copies ofindexed by,and so the identification
is a special case of ageneral resultrelating direct sums (ofmodules) to direct products.
If a vector space is not finite-dimensional, then its (algebraic) dual space isalwaysof 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 beisomorphicto the original vector space even if the latter is infinite-dimensional.
The proof of this inequality between dimensions results from the following.
Ifis an infinite-dimensional-vector space, the arithmetical properties ofcardinal numbersimplies that
where cardinalities are denoted asabsolute values.For proving thatit suffices to prove thatwhich can be done with an argument similar toCantor's diagonal argument.[9]The exact dimension of the dual is given by theErdős–Kaplansky theorem.
Bilinear products and dual spaces
[edit]IfVis finite-dimensional, thenVis isomorphic toV∗.But there is in general nonatural isomorphismbetween these two spaces.[10]Anybilinear form⟨·,·⟩onVgives a mapping ofVinto its dual space via
where the right hand side is defined as the functional onVtaking eachw∈Vto⟨v,w⟩.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 ofV∗. IfVis finite-dimensional, then this is an isomorphism onto all ofV∗.Conversely, any isomorphismfromVto a subspace ofV∗(resp., all ofV∗ifVis finite dimensional) defines a unique nondegenerate bilinear formonVby
Thus there is a one-to-one correspondence between isomorphisms ofVto a subspace of (resp., all of)V∗and nondegenerate bilinear forms onV.
If the vector spaceVis over thecomplexfield, then sometimes it is more natural to considersesquilinear formsinstead of bilinear forms. In that case, a given sesquilinear form⟨·,·⟩determines an isomorphism ofVwith thecomplex conjugateof the dual space
The conjugate of the dual spacecan be identified with the set of all additive complex-valued functionalsf:V→Csuch that
Injection into the double-dual
[edit]There is anaturalhomomorphismfrominto the double dual,defined byfor all.In other words, ifis the evaluation map defined by,thenis defined as the map.This mapis alwaysinjective;[nb 3]and it is always anisomorphismifis 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.
Transpose of a linear map
[edit]Iff:V→Wis alinear map,then thetranspose(ordual)f∗:W∗→V∗is defined by
for every.The resulting functionalinis called thepullbackofalong.
The following identity holds for alland:
where the bracket [·,·] on the left is the natural pairing ofVwith its dual space, and that on the right is the natural pairing ofWwith its dual. This identity characterizes the transpose,[12]and is formally similar to the definition of theadjoint.
The assignmentf↦f∗produces aninjectivelinear map between the space of linear operators fromVtoWand the space of linear operators fromW∗toV∗;this homomorphism is anisomorphismif and only ifWis finite-dimensional. IfV=Wthen the space of linear maps is actually analgebraundercomposition of maps,and the assignment is then anantihomomorphismof algebras, meaning that(fg)∗=g∗f∗. In the language ofcategory theory,taking the dual of vector spaces and the transpose of linear maps is therefore acontravariant functorfrom the category of vector spaces overFto itself. It is possible to identify (f∗)∗withfusing the natural injection into the double dual.
If the linear mapfis represented by thematrixAwith respect to two bases ofVandW,thenf∗is represented by thetransposematrixATwith respect to the dual bases ofW∗andV∗,hence the name. Alternatively, asfis represented byAacting 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 onRn,which identifies the space of column vectors with the dual space of row vectors.
Quotient spaces and annihilators
[edit]Letbe a subset of. Theannihilatorofin,denoted here,is the collection of linear functionalssuch thatfor all. That is,consists of all linear functionalssuch that the restriction tovanishes:. 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 ofreverses inclusions, so that if,then
Ifandare two subsets ofthen
Ifis any family of subsets ofindexed bybelonging to some index set,then
In particular ifandare subspaces ofthen
and[nb 3]
Ifis finite-dimensional andis avector subspace,then
after identifyingwith its image in the second dual space under the double duality isomorphism.In particular, forming the annihilator is aGalois connectionon the lattice of subsets of a finite-dimensional vector space.
Ifis a subspace ofthen thequotient spaceis a vector space in its own right, and so has a dual. By thefirst isomorphism theorem,a functionalfactors throughif and only ifis in thekernelof.There is thus an isomorphism
As a particular consequence, ifis adirect sumof two subspacesand,thenis a direct sum ofand.
Dimensional analysis
[edit]The dual space is analogous to a "negative" -dimensional space. Most simply, since a vectorcan be paired with a covectorby the natural pairing to obtain a scalar, a covector can "cancel" the dimension of a vector, similar toreducing a fraction.Thus while the direct sumis a-dimensional space (ifis-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:occurrencesperunit 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.
Continuous dual space
[edit]When dealing withtopological vector spaces,thecontinuouslinear 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-dimensionalnormed 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 spacesthe terms "continuous dual space" and "topological dual space" are often replaced by "dual space".
For atopological vector spaceitscontinuous 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 functionsand its dualthe space of arbitrarydistributions(generalized functions); the space of arbitrary test functionsand its dualthe space of compactly supported distributions; and the space of rapidly decreasing test functionstheSchwartz space,and its dualthe space oftempered distributions(slowly growing distributions) in the theory ofgeneralized functions.
Properties
[edit]IfXis aHausdorfftopological vector space(TVS), then the continuous dual space ofXis identical to the continuous dual space of thecompletionofX.[1]
Topologies on the dual
[edit]There is a standard construction for introducing a topology on the continuous dualof a topological vector space.Fix a collectionofbounded subsetsof. This gives the topology onof uniform convergence on sets fromor what is the same thing, the topology generated byseminormsof the form
whereis a continuous linear functional on,andruns over the class
This means that a net of functionalstends to a functionalinif and only if
Usually (but not necessarily) the classis supposed to satisfy the following conditions:
- Each pointofbelongs to some set:
- Each two setsandare contained in some set:
- is closed under the operation of multiplication by scalars:
If these requirements are fulfilled then the corresponding topology onis Hausdorff and the sets
form its local base.
Here are the three most important special cases.
- Thestrong topologyonis the topology of uniform convergence onbounded subsetsin(so herecan be chosen as the class of all bounded subsets in).
Ifis anormed vector space(for example, aBanach spaceor aHilbert space) then the strong topology onis normed (in fact a Banach space if the field of scalars is complete), with the norm
- Thestereotype topologyonis the topology of uniform convergence ontotally bounded setsin(so herecan be chosen as the class of all totally bounded subsets in).
- Theweak topologyonis the topology of uniform convergence on finite subsets in(so herecan be chosen as the class of all finite subsets in).
Each of these three choices of topology onleads to a variant ofreflexivity propertyfor topological vector spaces:
- Ifis endowed with thestrong topology,then the corresponding notion of reflexivity is the standard one: the spaces reflexive in this sense are just calledreflexive.[18]
- Ifis 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.
- Ifis 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]
Examples
[edit]Let 1 <p< ∞ be a real number and consider the Banach spaceℓ pof allsequencesa= (an)for which
Define the numberqby1/p+ 1/q= 1.Then the continuous dual ofℓ pis naturally identified withℓ q:given an element,the corresponding element ofℓ qis the sequencewheredenotes the sequence whosen-th term is 1 and all others are zero. Conversely, given an elementa= (an) ∈ℓ q,the corresponding continuous linear functionalonℓ pis defined by
for allb= (bn) ∈ℓ p(seeHölder's inequality).
In a similar manner, the continuous dual ofℓ 1is naturally identified withℓ ∞(the space of bounded sequences). Furthermore, the continuous duals of the Banach spacesc(consisting of allconvergentsequences, with thesupremum norm) andc0(the sequences converging to zero) are both naturally identified withℓ 1.
By theRiesz representation theorem,the continuous dual of a Hilbert space is again a Hilbert space which isanti-isomorphicto the original space. This gives rise to thebra–ket notationused by physicists in the mathematical formulation ofquantum mechanics.
By theRiesz–Markov–Kakutani representation theorem,the continuous dual of certain spaces of continuous functions can be described using measures.
Transpose of a continuous linear map
[edit]IfT:V → Wis a continuous linear map between two topological vector spaces, then the (continuous) transposeT′:W′ → V′is defined by the same formula as before:
The resulting functionalT′(φ)is inV′.The assignmentT → T′produces a linear map between the space of continuous linear maps fromVtoWand the space of linear maps fromW′toV′. WhenTandUare composable continuous linear maps, then
WhenVandWare normed spaces, the norm of the transpose inL(W′,V′)is equal to that ofTinL(V,W). Several properties of transposition depend upon theHahn–Banach theorem. For example, the bounded linear mapThas dense range if and only if the transposeT′is injective.
WhenTis acompactlinear map between two Banach spacesVandW,then the transposeT′is compact. This can be proved using theArzelà–Ascoli theorem.
WhenVis a Hilbert space, there is an antilinear isomorphismiVfromVonto its continuous dualV′. For every bounded linear mapTonV,the transpose and theadjointoperators are linked by
WhenTis a continuous linear map between two topological vector spacesVandW,then the transposeT′is continuous whenW′andV′are equipped with "compatible" topologies: for example, when forX=VandX=W,both dualsX′have thestrong topologyβ(X′,X)of uniform convergence on bounded sets ofX,or both have the weak-∗ topologyσ(X′,X)of pointwise convergence onX. The transposeT′is continuous fromβ(W′,W)toβ(V′,V),or fromσ(W′,W)toσ(V′,V).
Annihilators
[edit]Assume thatWis a closed linear subspace of a normed spaceV,and consider the annihilator ofWinV′,
Then, the dual of the quotientV / W can be identified withW⊥,and the dual ofWcan be identified with the quotientV′ / W⊥.[21] Indeed, letPdenote the canonicalsurjectionfromVonto the quotientV / W ;then, the transposeP′is an isometric isomorphism from(V / W )′intoV′,with range equal toW⊥. Ifjdenotes the injection map fromWintoV,then the kernel of the transposej′is the annihilator ofW:
and it follows from theHahn–Banach theoremthatj′induces an isometric isomorphism V′ / W⊥→W′.
Further properties
[edit]If the dual of a normed spaceVisseparable,then so is the spaceVitself. The converse is not true: for example, the spaceℓ 1is separable, but its dualℓ ∞is not.
Double dual
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Double_dual_nature.svg/220px-Double_dual_nature.svg.png)
In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operatorΨ:V→V′′from a normed spaceVinto its continuous double dualV′′,defined by
As a consequence of theHahn–Banach theorem,this map is in fact anisometry,meaning‖ Ψ(x) ‖ = ‖x‖for allx∈V. Normed spaces for which the map Ψ is abijectionare calledreflexive.
WhenVis atopological vector spacethen Ψ(x) can still be defined by the same formula, for everyx∈V,however several difficulties arise. First, whenVis notlocally convex,the continuous dual may be equal to { 0 } and the map Ψ trivial. However, ifVisHausdorffand locally convex, the map Ψ is injective fromVto the algebraic dualV′∗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 dualV′,so that the continuous double dualV′′is not uniquely defined as a set. Saying that Ψ maps fromVtoV′′,or in other words, that Ψ(x) is continuous onV′for everyx∈V,is a reasonable minimal requirement on the topology ofV′,namely that the evaluation mappings
be continuous for the chosen topology onV′.Further, there is still a choice of a topology onV′′,and continuity of Ψ depends upon this choice. As a consequence, definingreflexivityin this framework is more involved than in the normed case.
See also
[edit]- Covariance and contravariance of vectors
- Dual module
- Dual norm
- Duality (mathematics)
- Duality (projective geometry)
- Pontryagin duality
- Reciprocal lattice– dual space basis, in crystallography
Notes
[edit]- ^Forused in this way, seeAn Introduction to Manifolds(Tu 2011,p. 19). This notation is sometimes used whenis reserved for some other meaning. For instance, in the above text,is frequently used to denote the codifferential of,so thatrepresents the pullback of the form. Halmos (1974,p. 20) usesto denote the algebraic dual of.However, other authors usefor the continuous dual, while reservingfor the algebraic dual (Trèves 2006,p. 35).
- ^In many areas, such asquantum mechanics,⟨·,·⟩is reserved for asesquilinear formdefined onV×V.
- ^abcdSeveral assertions in this article require theaxiom of choicefor 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 thathas a basis. It is also needed to show that the dual of an infinite-dimensional vector spaceis nonzero, and hence that the natural map fromto its double dual is injective.
- ^To be precise, continuous Fourier analysis studies the space offunctionalswith domain a vector space and the space of functionals on the dual vector space.
- ^IfVis locally convex but not Hausdorff, thekernelof Ψ is the smallest closed subspace containing {0}.
References
[edit]- ^abNarici & Beckenstein 2011,pp. 225–273.
- ^Katznelson & Katznelson (2008)p. 37, §2.1.3
- ^abTu (2011)p. 19, §3.1
- ^Axler (2015)p. 101, §3.94
- ^Halmos (1974)p. 20, §13
- ^Halmos (1974)p. 21, §14
- ^Misner, Thorne & Wheeler 1973
- ^Misner, Thorne & Wheeler 1973,§2.5
- ^Nicolas Bourbaki (1974). Hermann (ed.).Elements of mathematics: Algebra I, Chapters 1 - 3.p. 400.ISBN0201006391.
- ^Mac Lane & Birkhoff 1999,§VI.4
- ^Halmos (1974)pp. 25, 28
- ^Halmos (1974)§44
- ^Tao, Terence(2012-12-29)."A mathematical formalisation of dimensional analysis".
Similarly, one can defineas the dual space to...
- ^abRobertson & Robertson 1964,II.2
- ^abSchaefer 1966,II.4
- ^Rudin 1973,3.1
- ^Bourbaki 2003,II.42
- ^Schaefer 1966,IV.5.5
- ^Schaefer 1966,IV.1
- ^Schaefer 1966,IV.1.2
- ^Rudin 1991,chapter 4
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