Duality is a very pervasive and important concept in (modern) mathematics. See Encyclopedia of Mathematics for a list.
Whenever we have two mathematical objects A and B, a set F of "scalars" of some kind, and a function $\beta: A \times B \to F$ 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 $\beta$ are called pairings. {III.19 Gowers}
Given two distinct sets of mathematical objects, duality means they are isomorphic: $A \cong B$, or equivalently, $\exists f: A \to B, f^{-1}: B \to A$. The mathematical structure of the objects are preserved; all propositions that hold in one formulation also hold in the dual formulation.
When the two sets are identical, the duality (isomorphism) $* : A \to A$ is often (but not always) an involution (对合): $x^{ ** } = x, \forall x \in A$. In this case, an element is self-dual to a dual operation if it is a fixed point of the 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, which is the major motivation of studying dual objects. Understood more broadly, every Math and Formal Theory is a candidate dual representation of (certain aspects of) the physical world. When a duality is properly established, a formal theory can help us understand the world with parsimony.
A category (范畴) $\mathcal{C} = (\text{Ob}\mathcal{C}, \text{Mor}\mathcal{C}, \circ)$ consists of a class of objects (物件), a class of morphisms (态射) from one object to another, and a composition (复合) operator on compatible pairs of morphisms, which satisfies:
A category $\mathcal{C}$ is said to be small if $\text{Ob}\mathcal{C}$ and $\text{Mor}\mathcal{C}$ are sets. Given two objects $A,B \in \text{Ob}\mathcal{C}$, their set of morphisms $H(A,B)$ in category $\mathcal{C}$ is defined such that $\alpha \in H(A,B) \iff \alpha: A \to B, \alpha \in \text{Mor}\mathcal{C}$. A morphism $f: A \to B$ is an isomorphism (同构) if it has an inverse: $\exists f^{-1}: B \to A$, $f \circ f^{-1} = 1_B, f^{-1} \circ f = 1_A$. Two objects are isomorphic if there is an isomorphism between them.
In most concrete categories over sets, an object is some mathematical structure; a morphism is a map between two objects; and the composition is just function composition.
Logic:
Bi-implication is self-dual.
Set theory:
In projective geometry, line and point are dual concepts. In three-dimensional projective geometry, point and plane are dual, while line is self-dual. 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".
Differential geometry: k-form and k-dimensional surface.
Linear algebra, only considering finite dimensional vector spaces:
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. One kind of problem (existence; surjection) is converted into a different kind (uniqueness; injection) in the dual formulation.
Abstract Algebra: Pontryagin duality, Abelian groups;
In topology, closed and open are dual concepts.
(closed convex set, polar set)
Structure: linearity, topology, etc.
Manifold: Poincare duality
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}
Functional analysis duality pairing: $\langle , \rangle$. Some properties of a function are naturally expressed in the dual formulation.
Fourier transform $\mathcal{F}$ on absolutely integrable continuous functions with real domain and complex range. 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.
Banach space:
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.
edge and vertex (undirected graph and its line graph)