In set theory the theorems remain true if one replaces the sets by their complements and interchanges union and intersection. There is a duality between sets and their complements.

The set of all linear functionals of a vectors space is a dual space to the vector space. If we have a Banach space then the space of all continuous linear functionals is the dual space. The duality induces pairings.

Polar bodies are defined using the inner product on Euclidean n space. A Platonic solid centered at the origin has a dual of polar bodies which are multiples of the dual Platonic solid.