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Definition

Let $G$ be a commutative (Hausdorff) topological group. A (continuous) character? of $G$ is any continuous homomorphism $\chi :G\to S^1$. The Pontrjagin dual group $\hat{G}$ is the commutative group of all characters of $G$ with pointwise multiplication (that is multiplication induced by multiplication in the circle group, the multiplication of norm-$1$ complex numbers in $S^1\subset ℂ$) and with the topology of uniform convergence? on each compact $K\subset G$ (this is equivalent to the compact-open topology).

For example, the Pontrjagin dual of the additive group of integers $\mathbb{Z}$ is the circle group $S^1$, and conversely, $\mathbb{Z}$ is the Pontrjagin dual of $S^1$. This pairing of dual topological groups, given by $(n,z)\mapsto z^n$, is related to the subject of Fourier series. In general, the dual of a discrete group is a compact group and conversely. The group $\widehat{ℝ}$ is isomorphic again to $ℝ$ (the additive group of real numbers), with the pairing given by $(x,p)\mapsto {\mathrm{e}}^{\mathrm{i}xp}$; similarly, $\widehat{{ℝ}^n}$ is isomorphic to the Cartesian space ${ℝ}^n$.

Pontrjagin duality theorem

Thus, the functor

is an equivalence, in fact an adjoint equivalence whose unit

and whose counit (the same arrow read in the opposite category) are isomorphisms. This contravariant self-equivalence restricts to equivalences

where $\mathrm{Ab}$ is the category of (discrete topological) groups and $\mathrm{CompAb}$ is the category of compact Hausdorff topological abelian groups, each embedded in $\mathrm{LocCompAb}$ in the evident way.

The Fourier transform on locally compact abelian groups is formulated in terms of Pontrjagin duals (see below).

There is a recent categorification of the Pontrjagin duality theorem by U. Bunke and T. Schick, motivated by applications to topological T-duality.

Applications

Pontrjagin duality underlies the abstract framework of Fourier analysis? on locally compact Hausdorff abelian groups $G$: by Fourier duality? on $G$, there is a Hilbert space isomorphism (Fourier transform)

where $d\mu$ is a suitable choice of Haar measure on $G$, and $d\hat{\mu }$ is a suitable choice of Haar measure on the dual group. Fourier duality is compatible with Pontrjagin duality in the sense that if $\hat{\hat{G}}$ is identified with $G$, then ${ℱ}_{\hat{G}}$ is the inverse of ${ℱ}_G$.