Dual space


In mathematics, any vector space has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on together with the vector space structure of pointwise 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 algebraic dual space. When defined for a topological vector space, 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 analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis.

Early terms for dual include polarer Raum [Hahn 1927], espace conjugué, adjoint space [Alaoglu 1940], and transponierter Raum [Schauder 1930] and [Banach 1932]. The term dual is due to Bourbaki 1938.[1]

Algebraic dual space edit

Given any vector space   over a field  , the (algebraic) dual space  [2] (alternatively denoted by  [3] or  [4][5])[nb 1] is defined as the set of all linear maps   (linear functionals). Since linear maps are vector space homomorphisms, 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  .

Elements of the algebraic dual space   are sometimes called covectors, one-forms, or linear 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 nondegenerate bilinear mapping[nb 2]   called the natural pairing.

Finite-dimensional case edit

If   is finite-dimensional, then   has the same dimension as  . Given a basis   in  , it is possible to construct a specific basis in  , called the dual basis. This dual basis is a set   of 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


where   is the Kronecker delta symbol. This property is referred to as the bi-orthogonality property.


Consider   the basis of V. Let   be defined as the following:


We have:

  1.   are linear functionals. Indeed, for   such as   and   (i.e.,   and  ). Then,   and  . Therefore,   for  .
  2. Suppose  . Applying this functional on the basis vectors of   successively, lead us to   (The functional applied in   results in  ). Therefore,   is linearly independent on  .
  3. Lastly, consider  . Then

and   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   are one-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  . 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, then


where   is the identity 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 of   real numbers, its dual space is typically written as the space of rows of   real numbers. Such a row acts on   as a linear functional by ordinary matrix 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 geometrical vectors in the plane, then the level 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 the level 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]

Infinite-dimensional case edit

If   is not finite-dimensional but has a basis[nb 3]   indexed by an infinite set  , then the same construction as in the finite-dimensional case yields linearly independent elements   ( ) of the dual space, but they will not form a basis.

For instance, consider the space  , whose elements are those sequences 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 of all 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  . The dimension of   is countably 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   of all 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 the direct 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), the direct product of infinitely many copies of   indexed by  , and so the identification


is a special case of a general result relating direct sums (of modules) to direct products.

If a vector space is not finite-dimensional, then its (algebraic) dual space is always of larger dimension (as a cardinal number) than the original vector space. This is in contrast to the case of the continuous dual space, discussed below, which may be isomorphic 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 of cardinal numbers implies that


where cardinalities are denoted as absolute values. For proving that   it suffices to prove that   which can be done with an argument similar to Cantor's diagonal argument.[citation needed] The exact dimension of the dual is given by the Erdős–Kaplansky theorem.

Bilinear products and dual spaces edit

If V is finite-dimensional, then V is isomorphic to V. But there is in general no natural isomorphism between these two spaces.[9] Any bilinear form ⟨·,·⟩ on V gives a mapping of V into its dual space via


where the right hand side is defined as the functional on V taking each wV to v, w. In other words, the bilinear form determines a linear mapping


defined by


If the bilinear form is nondegenerate, 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   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


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 field, then sometimes it is more natural to consider sesquilinear forms instead of bilinear forms. In that case, a given sesquilinear form ⟨·,·⟩ determines an isomorphism of V with the complex conjugate of the dual space


The conjugate of the dual space   can be identified with the set of all additive complex-valued functionals f : VC such that


Injection into the double-dual edit

There is a natural homomorphism   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 always injective;[nb 3] and it is always an isomorphism if   is finite-dimensional.[10] Indeed, the isomorphism of a finite-dimensional vector space with its double dual is an archetypal example of a natural 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

If f : VW is a linear map, then the transpose (or dual) f : WV is defined by


for every  . The resulting functional   in   is called the pullback of   along  .

The following identity holds for all   and  :


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,[11] and is formally similar to the definition of the adjoint.

The assignment ff produces an injective 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 if and only if W is finite-dimensional. If V = W then the space of linear maps is actually an algebra under composition of maps, and the assignment is then an antihomomorphism of algebras, meaning that (fg) = gf. In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor 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 A with respect to two bases of V and W, then f is represented by the transpose matrix AT 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 Rn, which identifies the space of column vectors with the dual space of row vectors.

Quotient spaces and annihilators edit

Let   be a subset of  . The annihilator 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) the orthogonal 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   then


If   is any family of subsets of   indexed by   belonging to some index set  , then


In particular if   and   are subspaces of   then


and[nb 3]


If   is finite-dimensional and   is a vector subspace, then


after identifying   with its image in the second dual space under the double duality isomorphism  . In particular, forming the annihilator is a Galois connection on the lattice of subsets of a finite-dimensional vector space.

If   is a subspace of   then the quotient space   is a vector space in its own right, and so has a dual. By the first isomorphism theorem, a functional   factors through   if and only if   is in the kernel of  . There is thus an isomorphism


As a particular consequence, if   is a direct sum of two subspaces   and  , then   is a direct sum of   and  .

Dimensional analysis edit

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 to reducing 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 by tensor contraction.

This arises in physics via dimensional analysis, where the dual space has inverse units.[12] Under the natural pairing, these units cancel, and the resulting scalar value is dimensionless, as expected. For example, in (continuous) Fourier analysis, or more broadly time–frequency analysis:[nb 4] given a one-dimensional vector space with a unit of time  , the dual space has units of frequency: occurrences per unit of time (units of  ). For example, if time is measured in seconds, the corresponding dual unit is the inverse 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 measures inverse length.

Continuous dual space edit

When dealing with topological vector spaces, the continuous 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 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 maps. Nevertheless, in the theory of topological vector spaces the terms "continuous dual space" and "topological dual space" are often replaced by "dual space".

For a topological vector space   its continuous dual space,[13] or topological dual space,[14] or just dual space[13][14][15][16] (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 supported test functions   and its dual   the space of arbitrary distributions (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   the Schwartz space, and its dual   the space of tempered distributions (slowly growing distributions) in the theory of generalized functions.

Properties edit

If X is a Hausdorff topological vector space (TVS), then the continuous dual space of X is identical to the continuous dual space of the completion of X.[1]

Topologies on the dual edit

There is a standard construction for introducing a topology on the continuous dual   of a topological vector space  . Fix a collection   of bounded subsets of  . This gives the topology on   of uniform convergence on sets from   or what is the same thing, the topology generated by seminorms 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.

  • The strong topology on   is the topology of uniform convergence on bounded subsets in   (so here   can be chosen as the class of all bounded subsets in  ).

If   is a normed vector space (for example, a Banach space or a Hilbert space) then the strong topology on   is normed (in fact a Banach space if the field of scalars is complete), with the norm

  • The stereotype topology on   is the topology of uniform convergence on totally bounded sets in   (so here   can be chosen as the class of all totally bounded subsets in  ).
  • The weak 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 of reflexivity property for topological vector spaces:

  • If   is endowed with the strong topology, then the corresponding notion of reflexivity is the standard one: the spaces reflexive in this sense are just called reflexive.[17]
  • If   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   is endowed with the weak topology, then the corresponding reflexivity is presented in the theory of dual pairs:[18] the spaces reflexive in this sense are arbitrary (Hausdorff) locally convex spaces with the weak topology.[19]

Examples edit

Let 1 < p < ∞ be a real number and consider the Banach space  p of all sequences a = (an) for which


Define the number q by 1/p + 1/q = 1. Then the continuous dual of p is naturally identified with q: given an element  , the corresponding element of q is the sequence   where   denotes the sequence whose n-th term is 1 and all others are zero. Conversely, given an element a = (an) ∈ q, the corresponding continuous linear functional   on p is defined by


for all b = (bn) ∈ p (see Hölder's inequality).

In a similar manner, the continuous dual of  1 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) and c0 (the sequences converging to zero) are both naturally identified with  1.

By the Riesz representation theorem, 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 used by physicists in the mathematical formulation of quantum mechanics.

By the Riesz–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

If T : V → W is a continuous linear map between two topological vector spaces, then the (continuous) transpose T′ : W′ → V′ is defined by the same formula as before:


The resulting functional T′(φ) is in V′. The assignment T → T′ produces a linear map between the space of continuous linear maps from V to W and the space of linear maps from W′ to V′. When T and U are composable continuous linear maps, then


When V and W are normed spaces, the norm of the transpose in L(W′, V′) is equal to that of T in L(V, W). Several properties of transposition depend upon the Hahn–Banach theorem. For example, the bounded linear map T has dense range if and only if the transpose T′ is injective.

When T is a compact linear map between two Banach spaces V and W, then the transpose T′ is compact. This can be proved using the Arzelà–Ascoli theorem.

When V is a Hilbert space, there is an antilinear isomorphism iV from V onto its continuous dual V′. For every bounded linear map T on V, the transpose and the adjoint operators are linked by


When T is a continuous linear map between two topological vector spaces V and W, then the transpose T′ is continuous when W′ and V′ are equipped with "compatible" topologies: for example, when for X = V and X = W, both duals X′ have the strong topology β(X′, X) of uniform convergence on bounded sets of X, or both have the weak-∗ topology σ(X′, X) of pointwise convergence on X. The transpose T′ is continuous from β(W′, W) to β(V′, V), or from σ(W′, W) to σ(V′, V).

Annihilators edit

Assume that W is a closed linear subspace of a normed space V, and consider the annihilator of W in V′,


Then, the dual of the quotient V / W can be identified with W, and the dual of W can be identified with the quotient V′ / W.[20] Indeed, let P denote the canonical surjection from V onto the quotient V / W; then, the transpose P′ is an isometric isomorphism from (V / W )′ into V′, with range equal to W. If j denotes the injection map from W into V, then the kernel of the transpose j′ is the annihilator of W:


and it follows from the Hahn–Banach theorem that j′ induces an isometric isomorphism V′ / WW′.

Further properties edit

If the dual of a normed space V is separable, then so is the space V itself. The converse is not true: for example, the space  1 is separable, but its dual  ∞ is not.

Double dual edit

This is a natural transformation of vector addition from a vector space to its double dual. x1, x2 denotes the ordered pair of two vectors. The addition + sends x1 and x2 to x1 + x2. 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 Ψ : VV′′ from a normed space V into its continuous double dual V′′, defined by


As a consequence of the Hahn–Banach theorem, this map is in fact an isometry, meaning ‖ Ψ(x) ‖ = ‖ x for all xV. Normed spaces for which the map Ψ is a bijection are called reflexive.

When V is a topological vector space then Ψ(x) can still be defined by the same formula, for every xV, however several difficulties arise. First, when V is not locally convex, the continuous dual may be equal to { 0 } and the map Ψ trivial. However, if V is Hausdorff and locally convex, the map Ψ is injective from V to the algebraic dual V′ 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 V′, so that the continuous double dual V′′ is not uniquely defined as a set. Saying that Ψ maps from V to V′′, or in other words, that Ψ(x) is continuous on V′ for every xV, is a reasonable minimal requirement on the topology of V′, namely that the evaluation mappings


be continuous for the chosen topology on V′. Further, there is still a choice of a topology on V′′, 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 edit

Notes edit

  1. ^ For   used in this way, see An 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).
  2. ^ In many areas, such as quantum mechanics, ⟨·,·⟩ is reserved for a sesquilinear form defined on V × V.
  3. ^ a b c d Several assertions in this article require the axiom 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.
  4. ^ 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.
  5. ^ If V is locally convex but not Hausdorff, the kernel of Ψ is the smallest closed subspace containing {0}.

References edit

  1. ^ a b Narici & Beckenstein 2011, pp. 225–273.
  2. ^ Katznelson & Katznelson (2008) p. 37, §2.1.3
  3. ^ a b Tu (2011) p. 19, §3.1
  4. ^ Axler (2015) p. 101, §3.94
  5. ^ Halmos (1974) p. 20, §13
  6. ^ Halmos (1974) p. 21, §14
  7. ^ Misner, Thorne & Wheeler 1973
  8. ^ Misner, Thorne & Wheeler 1973, §2.5
  9. ^ Mac Lane & Birkhoff 1999, §VI.4
  10. ^ Halmos (1974) pp. 25, 28
  11. ^ Halmos (1974) §44
  12. ^ Tao, Terence (2012-12-29). "A mathematical formalisation of dimensional analysis". Similarly, one can define   as the dual space to   ...
  13. ^ a b Robertson & Robertson 1964, II.2
  14. ^ a b Schaefer 1966, II.4
  15. ^ Rudin 1973, 3.1
  16. ^ Bourbaki 2003, II.42
  17. ^ Schaefer 1966, IV.5.5
  18. ^ Schaefer 1966, IV.1
  19. ^ Schaefer 1966, IV.1.2
  20. ^ Rudin 1991, chapter 4

Bibliography edit

External links edit