III.19 Duality

Duality is an important general theme that has manifestations in almost every area of mathematics. Over and over again, it turns out that one can associate with a given mathematical object a related, “dual” object that helps one to understand the properties of the object one started with. Despite the importance of duality in mathematics, there is no single definition that covers all instances of the phenomenon. So let us look at a few examples and at some of the characteristic features that they exhibit.

1 Platonic Solids

Suppose you take a cube, draw points at the centers of each of its six faces, and let those points be the vertices of a new polyhedron. The polyhedron you get will be a regular octahedron. What happens if you repeat the process? If you draw a point at the center of each of the eight faces of the octahedron, you will find that these points are the eight vertices of a cube. We say that the cube and the octahedron are dual to one another. The same can be done for the other Platonic solids: the dodecahedron and the icosahedron are dual to one another, while the dual of a tetrahedron is again a tetrahedron.

The duality just described does more than just split up the five Platonic solids into three groups: it allows us to associate statements about a solid with statements about its dual. For instance, two faces of a dodecahedron are adjacent if they share an edge, and this is so if and only if the corresponding vertices of the dual icosahedron are linked by an edge. And for this reason there is also a correspondence between edges of the dodecahedron and edges of the icosahedron.

2 Points and Lines in the Projective Plane

There are several equivalent definitions of the PROJECTIVE PLANE [I.3 §6.7]. One, which we shall use here, is that it is the set of all lines in ³ that go through the origin. These lines we call the “points” of the projective plane. In order to visualize this set as a geometrical object and to make its “points” more point-like, it is helpful to associate each line through the origin with the pair of points in ³ at which it intersects the unit sphere: indeed, one can define the projective plane as the unit sphere with opposite points identified.

A typical “line” in the projective plane is the set of all “points” (that is, lines through the origin) that lie in some plane through the origin. This is associated with the great circle in which that plane intersects the unit sphere, once again with opposite points identified.

There is a natural association between lines and points in the projective plane: each point P is associated with the line L that consists of all points orthogonal to P, and each line L is associated with the single point P that is orthogonal to all points in L. For example, if P is the z-axis, then the associated projective line L is the set of all lines through the origin that lie in the xy-plane, and vice versa. This association has the following basic property: if a point P belongs to a line L, then the line associated with P contains the point associated with L.

This allows us to translate statements about points and lines into logically equivalent statements about lines and points. For example, three points are collinear (that is, they all lie in a line) if and only if the corresponding lines are concurrent (that is, there is some point that is contained in all of them). In general, once you have proved a theorem in projective geometry, you get another, dual, theorem for free (unless the dual theorem turns out to be the same as the original one).

3 Sets and Their Complements

Let X be a set. If A is any subset of X, then the complement of A, written A^C, is the set of all elements of X that do not belong to A. The complement of the complement of A is clearly A, so there is a kind of duality between sets and their complements. De Morgcin’s laws are the statements that (A ∩ B)^C = A^C ∪ B^C and (A ∪ B)^c = A^C ∩ B^C: they tell us that complementation “turns intersections into unions,” and vice versa. Notice that if we apply the first law to A^C and B^C, then we find that (A^C ∩ B^c)^c = A ∪ B. Taking complements of both sides of this equality gives us the second law.

Because of de Morgan’s laws, any identity involving unions and intersections remains true when you interchange them. For example, one useful identity is A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). Applying this to the complements of the sets and using de Morgan’s laws, it is straightforward to deduce the equally useful identity A∩(B∪C) = (A∩B)∪(A∩C).

4 Dual Vector Spaces

Let V be a VECTOR SPACE [I.3 §2.3], over , say. The dual space V^* is defined to be the set of all linear functionals on V: that is, linear maps from V to . It is not hard to define appropriate notions of addition and scalar multiplication and show that these make V^* into a vector space as well.

Suppose that T is a LINEAR MAP [I.3 §4.2] from a vector space V to a vector space W. If we are given an element w^* of the dual space W^*, then we can use T and w^* to create an element of V^* as follows: it is the map that takes v to the real number w*(Tv). This map, which is denoted by T^*w^*, is easily checked to be linear. The function T^* is itself a linear map, called the adjoint of T, and it takes elements of W^* to elements of V^*.

This is a typical feature of duality: a function f from object A to object B very often gives rise to a function g from the dual of B to the dual of A.

Suppose that T^* is a surjection. Then if v ≠ v’, we can find v^* such that v^* (v) ≠ v^* (v’), and then W^* ∈ W^* such that T^*w^* = v^*, so that T^*w^*(v) ≠ T^*w^*(v’), and therefore w^*(Tv) ≠ w^*(Tv’). This implies that Tv ≠ Tv’, which proves that T is an injection. We can also prove that if T^* is an injection, then T is a surjection. Indeed, if T is not a surjection, then TV is a proper subspace of W, which allows us to find a nonzero linear functional w^* such that w^*(Tv) = 0 for every v∈ V, and hence such that T^*w^* = 0, which contradicts the injectivity of T^*. If V and W are finite dimensional, then (T^*)^* = T, so in this case we find that T is an injection if and only if T^* is a surjection, and vice versa. Therefore, we can use duality to convert an existence problem into a uniqueness problem. This conversion of one kind of problem into a different kind is another characteristic and very useful feature of duality.

If a vector space has additional structure, the definition of the dual space may well change. For instance, if X is a real BANACH SPACE [III.62], then X^* is defined to be the space of all continuous linear functionals from X to , rather than the space of all linear functionals. This space is also a Banach space: the norm of a continuous linear functional f is defined to be sup {|f(x)| : x ∈ X, ||x|| ≤ 1}. If X is an explicit example of a Banach space (such as one of the spaces discussed in FUNCTION SPACES [III.29]), it can be extremely useful to have an explicit description of the dual space. That is, one would like to find an explicitly described Banach space Y and a way of associating with each nonzero element y of Y a nonzero continuous linear functional φ_y defined on X, in such a way that every continuous linear functional is equal to φ_y for some y ∈ Y.

From this perspective, it is more natural to regard X and Y as having the same status. We can reflect this in our notation by writing 〈x, y〉 instead of φ_y (x). If we do this, then we are drawing attention to the fact that the map 〈.,.〉, which takes the pair (x, y) to the real number 〈x, y〉, is a continuous bilinear map from X × Y to .

More generally, whenever we have two mathematical objects A and B, a set S of “scalars” of some kind, and a function β: A × B →. S that is a structure-preserving map in each variable separately, we can think of the elements of A as elements of the dual of B, and vice versa. Functions like β are called pairings.

5 Polar Bodies

Let X be a subset of ⁿ and let 〈.,.〉) be the standard INNER PRODUCT [III.37] on ⁿ. Then the polar of X, denoted X^°, is the set of all points y ∈ ⁿ such that 〈x, y〉 ≤ 1 for every x ∈ X. It is not hard to check that X^° is closed and convex, and that if X is closed and convex, then (X^°)^° = X. Furthermore, if n = 3 and X is a Platonic solid centered at the origin, then X^° is (a multiple of) the dual Platonic solid, and if X is the “unit ball” of a normed space (that is, the set of all points of norm at most 1), then X^° is (easily identified with) the unit ball of the dual space.

6 Duals of Abelian Groups

If G is an Abelian group, then a character on G is a homomorphism from G to the group of all complex numbers of modulus 1. Two characters can be multiplied together in an obvious way, and this multiplication makes the set of all characters on G into another Abelian group, called the dual group, of the group G. Again, if G has a topological structure, then one usually imposes an additional continuity condition.

An important example is when the group is itself . It is not hard to show that the continuous homomorphisms from to all have the form e^iθ → e^inθ for some integer n (which may be negative or zero). Thus, the dual of is (isomorphic to) .

This form of duality between groups is called Pontryagin duality. Note that there is an easily defined pairing between G and : given an element g ∈ G and a character ψ ∈ , we define 〈g,ψ〉 to be ψ(g).

Under suitable conditions, this pairing extends to functions defined on G and . For instance, if G and are finite, and f : G → and F : → , then we can define 〈f,F〉 to be the complex number |G|^-1 f(g)F(ψ). In general, one obtains a pairing between a complex HILBERT SPACE [III.37] of functions on G and a Hilbert space of functions on .

This extended pairing leads to another important duality. Given a function in the Hilbert space L² (), its Fourier transform is the function ∈ l₂() that is defined by the formula

The Fourier transform, which can be defined similarly for functions on other Abelian groups, is immensely useful in many areas of mathematics. (See, for example, FOURIER TRANSFORMS [III.27] and REPRESENTATION THEORY [IV.9].) By contrast with some of the previous examples, it is not always easy to translate a statement about a function f into an equivalent statement about its Fourier transform , but this is what gives the Fourier transform its power: if you wish to understand a function f defined on , then you can explore its properties by looking at both f and . Some properties will follow from facts that are naturally expressed in terms of f and others from facts that are naturally expressed in terms of . Thus, the Fourier transform “doubles one’s mathematical power.”

7 Homology and Cohomology

Let X be a compact n-dimensional MANIFOLD [I.3 §6.9]. If M and M’ are an i-dimensional submanifold and an (n - i)-dimensional submanifold of X, respectively, and if they are well-behaved and in sufficiently general position, then they will intersect in a finite set of points. If one assigns either 1 or -1 to each of these points in a natural way that takes account of how M and M’ intersect, then the sum of the numbers at the points is an invariant called the intersection number of M and M’. This number turns out to depend only on the HOMOLOGY CLASSES [IV.6 §4] Of M and M’. Thus, it defines a map from H_i(X) × H_n-i (X) to , where we write H_r (X) for the r th homology group of X. This map is a group homomorphism in each variable separately, and the resulting pairing leads to a notion of duality called poincaré duality, and ultimately to the modern theory of cohomology, which is dual to homology. As with some of our other examples, many concepts associated with homology have dual concepts: for example, in homology one has a boundary map, whereas in cohomology there is a coboundary map (in the opposite direction). Another example is that a continuous map from X to Y gives rise to a homomorphism from the homology group H_i(X) to the homology group H_i(Y), and also to a homomorphism from the cohomology group Hⁱ(Y) to the cohomology group Hⁱ(X).

8 Further Examples Discussed in This Book

The examples above are not even close to a complete list: even in this book there are several more. For instance, the article On DIFFERENTIAL FORMS [III.16] discusses a pairing, and hence a duality, between k-forms and k-dimensional surfaces. (The pairing is given by integrating the form over the surface.) The article on DISTRIBUTIONS [III.18] shows how to use duality to give rigorous definitions of function-like objects such as the Dirac delta function. The article on MIRROR SYMMETRY [IV.16] discusses an astonishing (and still largely conjectural) duality between CALABI-YAU MANIFOLDS [III.6] and so-called “mirror manifolds.” Often the mirror manifold is much easier to understand than the original manifold, so this duality, like the Fourier transform, makes certain calculations possible that would otherwise be unthinkable. And the article on REPRESENTATION THEORY [IV.9] discusses the “Langlands dual” of certain (non-Abelian) groups: a proper understanding of this duality would solve many major open problems.