---
tag: R68HNB5SVR
kind: concept
name: Dual pair
msc: [46A20]
sources:
  - url: https://en.wikipedia.org/wiki/Dual_system
same_as:
  wikidata: Q96376663
  wikipedia: Dual system
---

## Summary

A dual pair, or dual system, consists of two vector spaces and a [[bilinear form|0SQZMXB6GP]] between them that separates points on both sides.
It is the abstract framework of duality theory for [[locally convex spaces|B6GY2KJMWW]].
The basic example is a Hausdorff locally convex space $E$ paired with its [[continuous dual|4A82S5FVRT]] $E'$ by $\langle x, f \rangle = f(x)$.
Topologies such as the [[weak topology|P6KHVSTPZV]] and the [[Mackey topology|W18EY4E5TH]] are defined from a dual pair alone.

## Explanations

### L1 human

In a dual pair, each side can be used to test the other.
A vector $x$ is known once one knows all the numbers $\langle x, y \rangle$, and likewise for $y$.
The simplest case is $\R^n$ paired with itself by the dot product.
In functional analysis one pairs a space of functions with a space of measures or of other functions, for example $\langle f, g \rangle = \int f g$.