A mathematical object oftentimes can be associated with a related object that helps understand the properties of the primal object. Duality refers to this general theme of mathematics despite a unifying definition.

  • Duality: Two interchangeable sets of concepts or operations, such that all propositions holding in one formulation also hold in the dual formulation.
  • Dual operation: often (but not always) an involution (对合) operation, \( A^{**}=A \).
  • Self-dual: fixed point of a dual operation, \( X^* = X \)

Dual object itself does not carry any extra value than its primal object, but it may be much easier to understand, or make certain otherwise unthinkable calculations possible. This is the major motivation of studying dual objects.

Mathematical Logic

Logic:

  • operation: negation
  • pairs: <proposition, negation>; <conjunction, disjunction>; <material implication, converse implication>

Bi-implication is self-dual.

Set theory

  • operation: complement
  • pairs: <set, complement set>; <union, intersection>; <contained in, contains>

Geometry

Projective Geometry

Dual concepts: line and point.

Feature: In projective geometry, once you have proved a theorem you get a dual theorem for free. (Unless the theorem is self-dual or the dual is trivial.) A pair of dual theorems is “Two points determine a line” and “Two lines determine a point.”

In three-dimensional projective geometry, point and plane are dual, while line is self-dual.

Differential geometry

k-form and k-dimensional surface

Algebra

Linear Algebra

Only considering finite dimensional vector spaces.

  • operation: bilinear map from pairs of vector spaces to scalars. (e.g. inner product <,>: V×V→R)
  • pairs: <vector, linear functional>, <vector space, dual vector space>, <linear map, adjoint linear map>, <surjection, injection>, <closed convex set, polar>

Feature:

  1. A function from object A to object B very often gives rise to a function from the dual of B to the dual of A.
  2. One kind of problem (existence; surjection) is converted into a different kind (uniqueness; injection) in the dual formulation.

Abstract Algebra

Pontryagin duality

Abelian groups

Topology

closed and open.

Banach space

  • operation: continuous bilinear map <,): : X×Y→R
  • pairs: <element, continuous linear functional>, <Banach space, dual Banach space>

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. {III.19 Gowers}

[structure: linearity, topology, etc.]

Manifold

Poincare duality

  • operation: intersection number \(H_i(X) \times H_i(X) \rightarrow Z\)
  • pairs: <homology, cohomology>

Analysis

Functional Analysis

The importance of the duality concept in functional analysis relies chiefly in the possibility of relating properties of representations in one space to those in its dual, a space that can be shown to possess certain analytical regularity properties, such as closure, even if its domain space does not. {Red-horse, R.G., 2009}

Duality pairing: \( \langle , \rangle \)

Feature: Some properties of a function are naturally expressed in the dual formulation.

  • Algebraic dual of linear vector space: the set of all linear functionals defined on the space
  • Topological dual of linear topological vector space: the set of all continuous linear functionals defined on this space.
    • Also called phase space.
    • a topology is naturally induced in the dual space
    • weak topology in the original space: topology induced by continuous linear functionals
    • the dual of a normed space is a Banach space
    • the dual of a Hilbert space is itself a Hilbert space
    • weak integral (Pettis integral)
    • strong integral
  • Distribution and test function
  • Dual spaces in Fourier transform: time domain and frequency domain

It is not always easy to translate a statement about a function into an equivalent statement about its Fourier transform.

Optimization

In mathematical optimization theory, duality means that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem. The difference is called the duality gap.

von Neumann conjectured the duality theorem for linear optimization, realizing that two person zero sum matrix game (where minimax = maximin) was equivalent to linear programming.

Graph theory

edge and vertex (undirected graph and its line graph)

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