Dual space
In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, 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.
Algebraic dual space
Given any vector space V over a field F, the (algebraic) dual space V^{∗} (alternatively denoted by or )^{[1]} is defined as the set of all linear maps φ: V → F (linear functionals). Since linear maps are vector space homomorphisms, the dual space is also sometimes denoted by Hom(V, F). The dual space V^{∗} itself becomes a vector space over F when equipped with an addition and scalar multiplication satisfying:
for all φ and ψ ∈ V^{∗}, x ∈ V, and a ∈ F. Elements of the algebraic dual space V^{∗} are sometimes called covectors or one-forms.
The pairing of a functional φ in the dual space V^{∗} and an element x of V is sometimes denoted by a bracket: φ(x) = [x,φ] ^{[2]} or φ(x) = ⟨φ,x⟩.^{[3]} This pairing defines a nondegenerate bilinear mapping^{[4]} ⟨·,·⟩ : V^{∗} × V → F called the natural pairing.
Finite-dimensional case
If V is finite-dimensional, then V^{∗} has the same dimension as V. Given a basis {e_{1}, ..., e_{n}} in V, it is possible to construct a specific basis in V^{∗}, called the dual basis. This dual basis is a set {e^{1}, ..., e^{n}} of linear functionals on V, defined by the relation
for any choice of coefficients c^{i} ∈ F. 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 biorthogonality property.
For example, if V is R^{2}, let its basis be chosen as {e_{1} = (1/2, 1/2), e_{2} = (0, 1)}. Note that the basis vectors are not orthogonal to each other. Then, e^{1} and e^{2} are one-forms (functions that map a vector to a scalar) such that e^{1}(e_{1}) = 1, e^{1}(e_{2}) = 0, e^{2}(e_{1}) = 0, and e^{2}(e_{2}) = 1. (Note: The superscript here is the index, not an exponent). We can express this system of equations using matrix notation as
Solving this equation, we find the dual basis to be {e^{1} = (2, 0), e^{2} = (−1, 1)}. Recalling that e^{1} and e^{2} are functionals, we can rewrite them as e^{1}(x, y) = 2x and e^{2}(x, y) = −x + y. In general, when V is R^{n}, if E = (e_{1}, ..., e_{n}) is a matrix whose columns are the basis vectors and Ê = (e^{1}, ..., e^{n}) is a matrix whose columns are the dual basis vectors, then
where I_{n} is an identity matrix of order n. The biorthogonality property of these two basis sets allows us to represent any point x in V as
even when the basis vectors are not orthogonal to each other.
In particular, if we interpret R^{n} as the space of columns of n real numbers, its dual space is typically written as the space of rows of n real numbers. Such a row acts on R^{n} as a linear functional by ordinary matrix multiplication. One way to see this is that a functional maps every n-vector x into a real number y. Then, seeing this functional as a matrix M, and x, y as a n × 1 matrix and a 1 × 1 matrix (trivially, a real number) respectively, if we have Mx = y, then, by dimension reasons, M must be a 1 × n matrix, i.e., M must be a row vector.
If V consists of the space of geometrical vectors 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, one needs only to determine which of the lines the vector lies on. Or, informally, one "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.^{[5]}
Infinite-dimensional case
If V is not finite-dimensional but has a basis^{[6]} e_{α} indexed by an infinite set A, then the same construction as in the finite-dimensional case yields linearly independent elements e^{α} (α ∈ A) of the dual space, but they will not form a basis.
Consider, for instance, the space R^{∞}, 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 N: for i ∈ N, 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) R^{N}, the space of all sequences of real numbers: such a sequence (a_{n}) is applied to an element (x_{n}) of R^{∞} to give the number
which is a finite sum because there are only finitely many nonzero x_{n}. The dimension of R^{∞} is countably infinite, whereas R^{N} does not have a countable basis.
This observation generalizes to any^{[6]} infinite-dimensional vector space V over any field F: a choice of basis {e_{α} : α ∈ A} identifies V with the space (F^{A})_{0} of functions f : A → F such that f_{α} = f(α) is nonzero for only finitely many α ∈ A, where such a function f is identified with the vector
in V (the sum is finite by the assumption on f, and any v ∈ V may be written in this way by the definition of the basis).
The dual space of V may then be identified with the space F^{A} of all functions from A to F: a linear functional T on V is uniquely determined by the values θ_{α} = T(e_{α}) it takes on the basis of V, and any function θ : A → F (with θ(α) = θ_{α}) defines a linear functional T on V by
Again the sum is finite because f_{α} is nonzero for only finitely many α.
Note that (F^{A})_{0} may be identified (essentially by definition) with the direct sum of infinitely many copies of F (viewed as a 1-dimensional vector space over itself) indexed by A, i.e., there are linear isomorphisms
On the other hand, F^{A} is (again by definition), the direct product of infinitely many copies of F indexed by A, and so the identification
is a special case of a general result relating direct sums (of modules) to direct products.
Thus if the basis is infinite, then the 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.
Bilinear products and dual spaces
If V is finite-dimensional, then V is isomorphic to V^{∗}. But there is in general no natural isomorphism between these two spaces.^{[7]} 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 w ∈ V 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 subspaces 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 space V^{∗} can be identified with the set of all additive complex-valued functionals f: V → C such that
Injection into the double-dual
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 we define as the map . This map is always injective;^{[6]} it is an isomorphism if and only if is finite-dimensional.^{[8]} Indeed, the isomorphism of a finite-dimensional vector space with its double dual is an archetypal example of a natural isomorphism. Note that infinite-dimensional Hilbert spaces are not a counterexample to this, as they are isomorphic to their continuous duals, not to their algebraic duals.
Transpose of a linear map
If f : V → W is a linear map, then the transpose (or dual) f^{∗} : W^{∗} → V^{∗} is defined by
for every φ ∈ W^{∗}. The resulting functional f^{∗}(φ) in V^{∗} is called the pullback of φ along f.
The following identity holds for all φ ∈ W^{∗} and v ∈ 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,^{[9]} and is formally similar to the definition of the adjoint.
The assignment f ↦ f^{∗} 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)^{∗} = g^{∗}f^{∗}. 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. Note that one can 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 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^{0}, is the collection of linear functionals f ∈ V^{∗} such that [f, s] = 0 for all s ∈ S. That is, S^{0} consists of all linear functionals f : V → F such that the restriction to S vanishes: f|_{S} = 0.
The annihilator of a subset is itself a vector space. In particular, ∅^{0} = V^{∗} is all of V^{∗} (vacuously), whereas V^{0} = 0 is the zero subspace. Furthermore, the assignment of an annihilator to a subset of V reverses inclusions, so that if S ⊂ T ⊂ V, then
Moreover, if A and B are two subsets of V, then
and equality holds provided V is finite-dimensional. If A_{i} is any family of subsets of V indexed by i belonging to some index set I, then
In particular if A and B are subspaces of V, it follows that
If V is finite-dimensional, and W is a vector subspace, then
after identifying W with its image in the second dual space under the double duality isomorphism V ≈ V^{∗∗}. Thus, in particular, forming the annihilator is a Galois connection 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, a functional f : V → F factors through V/W if and only if W is in the kernel of f. There is thus an isomorphism
As a particular consequence, if V is a direct sum of two subspaces A and B, then V^{∗} is a direct sum of A^{0} and B^{0}.
Continuous dual space
When dealing with topological vector spaces, one is typically only interested in the continuous linear functionals from the space into the base field (or ). 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", since there is no serious need to consider discontinuous maps in this field.
For a topological vector space its continuous dual space,^{[10]} or topological dual space,^{[11]} or just dual space^{[10]}^{[11]}^{[12]}^{[13]} (in the sense of the theory of topological vector spaces) is defined as the space of all continuous linear functionals .
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 . Then one has 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 (e.g., 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.
Examples
Let 1 < p < ∞ be a real number and consider the Banach space ℓ^{ p} of all sequences a = (a_{n}) for which
Define the number q by 1/p + 1/q = 1. Then the continuous dual of ℓ^{ p} is naturally identified with ℓ^{ q}: given an element φ ∈ (ℓ^{ p})′, the corresponding element of ℓ^{ q} is the sequence (φ(e_{n})) where e_{n} denotes the sequence whose n-th term is 1 and all others are zero. Conversely, given an element a = (a_{n}) ∈ ℓ^{ q}, the corresponding continuous linear functional φ on ℓ^{ p} is defined by
for all b = (b_{n}) ∈ ℓ^{ p} (see Hölder's inequality).
In a similar manner, the continuous dual of ℓ^{ 1} is naturally identified with ℓ^{ ∞} (the space of bounded sequences). Furthermore, the continuous duals of the Banach spaces c (consisting of all convergent sequences, with the supremum norm) and c_{0} (the sequences converging to zero) are both naturally identified with ℓ^{ 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
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 i_{V} 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
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^{⊥}.^{[14]} 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′ / W^{⊥} → W′.
Further properties
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 ℓ^{ 1} is separable, but its dual ℓ^{ ∞} is not.
Topologies on the dual
The topology of V and the topology of real or complex numbers can be used to induce on V′ a dual space topology.
Double dual
In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator Ψ : V → V′′ 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 x in V. Normed spaces for which the map Ψ is a bijection are called reflexive.
When V is a topological vector space, one can still define Ψ(x) by the same formula, for every x ∈ V, 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.^{[15]}
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 x ∈ V, 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
- Duality (mathematics)
- Duality (projective geometry)
- Pontryagin duality
- Reciprocal lattice – dual space basis, in crystallography
- Covariance and contravariance of vectors
- Dual norm
- Dual module
Notes
- ^ For used in this way, see p. 19 of An Introduction to Manifolds (2nd edition, 2011, Springer) by Loring Tu. 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 F, so that represents the pullback of the form ω. Halmos's Finite-Dimensional Vector Spaces (2nd edition, 1958, Van Nostrand) uses to denote the algebraic dual of V ; see p. 20. However, other authors use for the continuous dual, while reserving for the algebraic dual.
- ^ Halmos (1974)
- ^ Misner, Thorne & Wheeler (1973)
- ^ In many areas, such as quantum mechanics, ⟨·,·⟩ is reserved for a sesquilinear form defined on V × V.
- ^ Misner, Thorne & Wheeler (1973, §2.5)
- ^ ^{a} ^{b} ^{c} 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 R^{N} 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.
- ^ MacLane & Birkhoff (1999, §VI.4)
- ^ Halmos, Paul R. (1958). Finite-Dimensional Vector Spaces (2nd Edition). Princeton, NJ: Van Nostrand. pp. 25, 28. ISBN 0-387-90093-4.
- ^ Halmos (1974, §44)
- ^ ^{a} ^{b} A.P.Robertson, W.Robertson (1964, II.2)
- ^ ^{a} ^{b} H.Schaefer (1966, II.4)
- ^ W.Rudin (1973, 3.1)
- ^ Nicolas Bourbaki (2003, II.42)
- ^ Rudin (1991, chapter 4)
- ^ If V is locally convex but not Hausdorff, the kernel of Ψ is the smallest closed subspace containing {0}.
References
- Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9
- Bourbaki, Nicolas. (2003), Elements of mathematics, Topological vector spaces, Springer-Verlag
- Halmos, Paul (1974), Finite-dimensional Vector Spaces, Springer, ISBN 0-387-90093-4
- Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001
- MacLane, Saunders; Birkhoff, Garrett (1999), Algebra (3rd ed.), AMS Chelsea Publishing, ISBN 0-8218-1646-2.
- Misner, Charles W.; Thorne, Kip S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, ISBN 0-7167-0344-0
- Rudin, Walter (1991). Functional analysis. McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5.
- Robertson, A.P.; Robertson, W. (1964). Topological vector spaces. Cambridge University Press.
- Schaefer, Helmuth H. (1971). Topological vector spaces. GTM. 3. New York: Springer-Verlag. ISBN 0-387-98726-6.