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.
for any choice of coefficients ci ∈ 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 In is an identity matrix of order n. The biorthogonality property of these two basis sets allows any point x ∈ V to be represented as
In particular, Rn can be interpreted 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 Rn as a linear functional by ordinary matrix multiplication. This is because 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 Mx = y then, by dimension reasons, M must be a 1 × n matrix; that is, 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, it suffices to determine which of the lines the vector lies on. Informally, this "counts" how many lines the vector crosses. More generally, if V is a vector space of any dimension, then the level sets of a linear functional in V∗ are parallel hyperplanes in V, and the action of a linear functional on a vector can be visualized in terms of these hyperplanes.
If V is not finite-dimensional but has a basis[nb 3] eα indexed by an infinite set A, then the same construction as in the finite-dimensional case yields linearly independent elements eα (α ∈ A) of the dual space, but they will not form a basis.
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, ei is the sequence consisting of all zeroes except in the i-th position, which is 1. The dual space of R∞ is (isomorphic to) RN, the space of all sequences of real numbers: each real sequence (an) defines a function where the element (xn) of R∞ is sent to the number
which is a finite sum because there are only finitely many nonzero xn. The dimension of R∞ is countably infinite, whereas RN does not have a countable basis.
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 FA of all functions from A to F: a linear functional T on V is uniquely determined by the values θα = T(eα) it takes on the basis of V, and any function θ : A → F (with θ(α) = θα) defines a linear functional T on V by
Again the sum is finite because fα is nonzero for only finitely many α.
The set (FA)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, FA 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.
It follows that, if a vector space is not finite-dimensional, then the axiom of choice implies that the algebraic dual space is always of larger dimension (as a cardinal number) than the original vector space (since, if two bases have the same cardinality, the spanned vector spaces have the same cardinality). This is in contrast to the case of the continuous dual space, discussed below, which may be isomorphic to the original vector space even if the latter is infinite-dimensional.
If V is finite-dimensional, then V is isomorphic to V∗. But there is in general no natural isomorphism between these two spaces. 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
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
where the bracket [·,·] on the left is the natural pairing of V with its dual space, and that on the right is the natural pairing of W with its dual. This identity characterizes the transpose, and is formally similar to the definition of the adjoint.
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. 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.
Let S be a subset of V. The annihilator of S in V∗, denoted here S0, is the collection of linear functionals f ∈ V∗ such that [f, s] = 0 for all s ∈ S. That is, S0 consists of all linear functionals f : V → F such that the restriction to S vanishes: f|S = 0. Within finite dimensional vector spaces, the annihilator is dual to (isomorphic to) the orthogonal complement.
and equality holds provided V is finite-dimensional. If Ai is any family of subsets of V indexed by i belonging to some index set I, then
after identifying W with its image in the second dual space under the double duality isomorphism V ≈ V∗∗. 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 A0 and B0.
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 c0 (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.
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 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).
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⊥. 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′.
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.
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
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.