III.19 Duality: Sets and Their Complements, Dual Vector Spaces and Polar Bodies: Page 188

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.

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