Duality (mathematics)

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In mathematics, a duality
translates concepts, theorems or mathematical structures into
other concepts, theorems or structures, in a one-to-one fashion,
often (but not always) by means of an involution operation:

if the dual of A is B, then the dual of B is A.
Such involutions sometimes have fixed points, so that the dual of A is A itself.

For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.

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In mathematical contexts, duality has numerous meanings.[1]
It has been described as "a very pervasive and important concept
in (modern) mathematics"[2] and "an important general theme that
has manifestations in almost every area of mathematics".[3]

Many mathematical dualities between objects of two types correspond to pairings,
bilinear functions from an object of one type and another object of the second type
to some family of scalars.

For instance, linear algebra duality corresponds in this way to bilinear maps from
pairs of vector spaces to scalars, the duality between distributions and
the associated test functions corresponds to the pairing in which
one integrates a distribution against a test function, and
Poincaré duality corresponds similarly to intersection number,
viewed as a pairing between submanifolds of a given manifold.[4]

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Duality is a very general and broad concept, without a strict definition that captures all those uses. When applied to specific concepts, there usually is a precise definition for just that context. The common idea is that there are two things which basically are just two sides of the same coin.

Common themes in this topic include:

Two different interpretations or descriptions of fundamentally the same structure or object
(e.g. roles of points and lines interchanged, roles of variables in LP changed)
Primal and dual often are the same kind of object
(e.g. incidence configuration, vector space, linear program, planar graph, …)
The dual of the dual is again the primal
Not every use of the word strictly satisfies all of these aspects, but the general idea usually is still the same.

https://math.stackexchange.com/quest...hat-is-duality

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

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In the theory of lattices, the dual lattice is a construction analogous to that of a dual vector space.
In certain respects, the geometry of the dual lattice of a lattice L is the reciprocal of the geometry of L,
a perspective which underlies many of its uses.

Dual lattices have many applications inside of lattice theory, theoretical computer science,
cryptography and mathematics more broadly.