Pontryagin dual

Builds on Locally compact abelian group Character of an abelian group Circle group Compact-open topology

Definition

Let $G$ be a locally compact abelian group, and let $\mathbb{T}$ be the circle group. The Pontryagin dual of $G$ is the set $\widehat{G}$ of all continuous characters $\chi \colon G \to \mathbb{T}$ , with:

the group operation $(\chi\psi)(x) = \chi(x)\psi(x)$ ;
the compact-open topology, that is, the topology of uniform convergence on compact subsets of $G$ .

Remarks

Example: for $G = \Z$ with the discrete topology, a character $\chi$ is fixed by $z = \chi(1) \in \mathbb{T}$ , since $\chi(n) = z^n$ , and each $z \in \mathbb{T}$ gives a character. Uniform convergence on finite sets is pointwise convergence, so $\widehat{\Z} \cong \mathbb{T}$ as topological groups. Non-example: the group of all characters of $\R$ , continuous or not, is not the Pontryagin dual of $\R$ . There are discontinuous homomorphisms $\R \to \mathbb{T}$ , built from a Hamel basis, and the dual excludes them. With the compact-open topology, $\widehat{G}$ is again a locally compact abelian group. If $G$ is compact, then $\widehat{G}$ is discrete, and if $G$ is discrete, then $\widehat{G}$ is compact.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`ET9QA03J3F`
URI	`m2://ET9QA03J3F`
Kind	definition
Writing	W0 stub
MSC	43A40, 22B05
Defines	Pontryagin dual
Uses	Locally compact abelian group, Character of an abelian group, Circle group, Compact-open topology

Pontryagin dual

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here