In mathematics, any vector space ${\displaystyle V}$ has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ${\displaystyle V}$, 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.

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

Given any vector space ${\displaystyle V}$ over a field ${\displaystyle F}$, the (algebraic) dual space ${\displaystyle V^{*))$[2] (alternatively denoted by ${\displaystyle V^{\lor ))$[3] or ${\displaystyle V'}$[4][5])[nb 1] is defined as the set of all linear maps ${\displaystyle \varphi :V\to F}$ (linear functionals). Since linear maps are vector space homomorphisms, the dual space may be denoted ${\displaystyle \hom(V,F)}$.[6] The dual space ${\displaystyle V^{*))$ itself becomes a vector space over ${\displaystyle F}$ when equipped with an addition and scalar multiplication satisfying:

Transpose of a linear map

 Main article: Transpose of a linear map

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

${\displaystyle f^{*}(\varphi )=\varphi \circ f\,}$

for every ${\displaystyle \varphi \in W^{*))$. The resulting functional ${\displaystyle f^{*}(\varphi )}$ in ${\displaystyle V^{*))$ is called the pullback of ${\displaystyle \varphi }$ along ${\displaystyle f}$.

The following identity holds for all ${\displaystyle \varphi \in W^{*))$ and ${\displaystyle v\in V}$:

${\displaystyle [f^{*}(\varphi ),\,v]=[\varphi ,\,f(v)],}$

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,[12] 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

Let S be a subset of V. The annihilator of S in V, denoted here S0, is the collection of linear functionals fV such that [f, s] = 0 for all sS. That is, S0 consists of all linear functionals f : VF 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.

The annihilator of a subset is itself a vector space. The annihilator of the zero vector is the whole dual space: ${\displaystyle \{0\}^{0}=V^{*))$, and the annihilator of the whole space is just the zero covector: ${\displaystyle V^{0}=\{0\}\subseteq V^{*))$. Furthermore, the assignment of an annihilator to a subset of V reverses inclusions, so that if STV, then

${\displaystyle 0\subseteq T^{0}\subseteq S^{0}\subseteq V^{*}.}$

If A and B are two subsets of V then

${\displaystyle A^{0}+B^{0}\subseteq (A\cap B)^{0},}$

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

${\displaystyle \left(\bigcup _{i\in I}A_{i}\right)^{0}=\bigcap _{i\in I}A_{i}^{0}.}$

In particular if A and B are subspaces of V then

${\displaystyle (A+B)^{0}=A^{0}\cap B^{0}.}$

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

${\displaystyle W^{00}=W}$

after identifying W with its image in the second dual space under the double duality isomorphism VV∗∗. 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 : VF factors through V/W if and only if W is in the kernel of f. There is thus an isomorphism

${\displaystyle (V/W)^{*}\cong W^{0}.}$

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.

Continuous dual space

When dealing with topological vector spaces, the continuous linear functionals from the space into the base field ${\displaystyle \mathbb {F} =\mathbb {C} }$ (or ${\displaystyle \mathbb {R} }$) 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 ${\displaystyle V^{*))$, denoted by ${\displaystyle V'}$. 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 ${\displaystyle V}$ 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) ${\displaystyle V'}$ is defined as the space of all continuous linear functionals ${\displaystyle \varphi :V\to {\mathbb {F} ))$.

Properties

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

 Main articles: Polar topology and Dual system

There is a standard construction for introducing a topology on the continuous dual ${\displaystyle V'}$ of a topological vector space ${\displaystyle V}$. Fix a collection ${\displaystyle {\mathcal {A))}$ of bounded subsets of ${\displaystyle V}$. This gives the topology on ${\displaystyle V}$ of uniform convergence on sets from ${\displaystyle {\mathcal {A)),}$ or what is the same thing, the topology generated by seminorms of the form

${\displaystyle \|\varphi \|_{A}=\sup _{x\in A}|\varphi (x)|,}$

where ${\displaystyle \varphi }$ is a continuous linear functional on ${\displaystyle V}$, and ${\displaystyle A}$ runs over the class ${\displaystyle {\mathcal {A)).}$

This means that a net of functionals ${\displaystyle \varphi _{i))$ tends to a functional ${\displaystyle \varphi }$ in ${\displaystyle V'}$ if and only if

${\displaystyle {\text{ for all ))A\in {\mathcal {A))\qquad \|\varphi _{i}-\varphi \|_{A}=\sup _{x\in A}|\varphi _{i}(x)-\varphi (x)|{\underset {i\to \infty }{\longrightarrow ))0.}$

Usually (but not necessarily) the class ${\displaystyle {\mathcal {A))}$ is supposed to satisfy the following conditions:

• Each point ${\displaystyle x}$ of ${\displaystyle V}$ belongs to some set ${\displaystyle A\in {\mathcal {A))}$:
${\displaystyle {\text{ for all ))x\in V\quad {\text{ there exists some ))A\in {\mathcal {A))\quad {\text{ such that ))x\in A.}$
• Each two sets ${\displaystyle A\in {\mathcal {A))}$ and ${\displaystyle B\in {\mathcal {A))}$ are contained in some set ${\displaystyle C\in {\mathcal {A))}$:
${\displaystyle {\text{ for all ))A,B\in {\mathcal {A))\quad {\text{ there exists some ))C\in {\mathcal {A))\quad {\text{ such that ))A\cup B\subseteq C.}$
• ${\displaystyle {\mathcal {A))}$ is closed under the operation of multiplication by scalars:
${\displaystyle {\text{ for all ))A\in {\mathcal {A))\quad {\text{ and all ))\lambda \in {\mathbb {F} }\quad {\text{ such that ))\lambda \cdot A\in {\mathcal {A)).}$

If these requirements are fulfilled then the corresponding topology on ${\displaystyle V'}$ is Hausdorff and the sets

${\displaystyle U_{A}~=~\left\{\varphi \in V'~:~\quad \|\varphi \|_{A}<1\right\},\qquad {\text{ for ))A\in {\mathcal {A))}$

form its local base.

Here are the three most important special cases.

• The strong topology on ${\displaystyle V'}$ is the topology of uniform convergence on bounded subsets in ${\displaystyle V}$ (so here ${\displaystyle {\mathcal {A))}$ can be chosen as the class of all bounded subsets in ${\displaystyle V}$).

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

${\displaystyle \|\varphi \|=\sup _{\|x\|\leq 1}|\varphi (x)|.}$
• The stereotype topology on ${\displaystyle V'}$ is the topology of uniform convergence on totally bounded sets in ${\displaystyle V}$ (so here ${\displaystyle {\mathcal {A))}$ can be chosen as the class of all totally bounded subsets in ${\displaystyle V}$).
• The weak topology on ${\displaystyle V'}$ is the topology of uniform convergence on finite subsets in ${\displaystyle V}$ (so here ${\displaystyle {\mathcal {A))}$ can be chosen as the class of all finite subsets in ${\displaystyle V}$).

Each of these three choices of topology on ${\displaystyle V'}$ leads to a variant of reflexivity property for topological vector spaces:

• If ${\displaystyle V'}$ 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 ${\displaystyle V'}$ 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 ${\displaystyle V'}$ 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

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

${\displaystyle \|\mathbf {a} \|_{p}=\left(\sum _{n=0}^{\infty }|a_{n}|^{p}\right)^{\frac {1}{p))<\infty .}$

Define the number q by 1/p + 1/q = 1. Then the continuous dual of p is naturally identified with q: given an element ${\displaystyle \varphi \in (\ell ^{p})'}$, the corresponding element of q is the sequence ${\displaystyle (\varphi ({\mathbf {e))_{n}))}$ where ${\displaystyle {\mathbf {e))_{n))$ 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 ${\displaystyle \varphi }$ on p is defined by

${\displaystyle \varphi (\mathbf {b} )=\sum _{n}a_{n}b_{n))$

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

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:

${\displaystyle T'(\varphi )=\varphi \circ T,\quad \varphi \in W'.}$

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

${\displaystyle (U\circ T)'=T'\circ U'.}$

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

${\displaystyle i_{V}\circ T^{*}=T'\circ i_{V}.}$

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′,

${\displaystyle W^{\perp }=\{\varphi \in V':W\subseteq \ker \varphi \}.}$

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:

${\displaystyle \ker(j')=W^{\perp ))$

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

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  1 is separable, but its dual  ∞ is not.

Double dual

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 ${\displaystyle [\Psi (x_{1})+'\Psi (x_{2})](\varphi )=\varphi (x_{1}+x_{2})=\varphi (x)}$ for any ${\displaystyle \varphi }$ 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

${\displaystyle \Psi (x)(\varphi )=\varphi (x),\quad x\in V,\ \varphi \in V'.}$

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 4]

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

${\displaystyle \varphi \in V'\mapsto \varphi (x),\quad x\in V,}$

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.

Notes

1. ^ For ${\displaystyle V^{\lor ))$ used in this way, see An Introduction to Manifolds (Tu 2011, p. 19). This notation is sometimes used when ${\displaystyle (\cdot )^{*))$ is reserved for some other meaning. For instance, in the above text, ${\displaystyle F^{*))$ is frequently used to denote the codifferential of ${\displaystyle F}$, so that ${\displaystyle F^{*}\omega }$ represents the pullback of the form ${\displaystyle \omega }$. Halmos (1974, p. 20) uses ${\displaystyle V'}$ to denote the algebraic dual of ${\displaystyle V}$. However, other authors use ${\displaystyle V'}$ for the continuous dual, while reserving ${\displaystyle V^{*))$ 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 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 RN 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.
4. ^ If V is locally convex but not Hausdorff, the kernel of Ψ is the smallest closed subspace containing {0}.

References

1. ^ a b Narici & Beckenstein 2011, pp. 225–273.
2. ^ Katznelson & Katznelson (2008) p. 37, §2.1.3
3. ^ Tu (2011) p. 19, §3.1
4. ^ Axler (2015) p. 101, §3.94
5. ^ Halmos (1974) p. 20, §13
6. ^ Tu (2011) p. 19, §3.1
7. ^ Halmos (1974) p. 21, §14
8. ^ Misner, Thorne & Wheeler 1973
9. ^
10. ^ Mac Lane & Birkhoff 1999, §VI.4
11. ^ Halmos (1974) pp. 25, 28
12. ^
13. ^ a b
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