Pontryagin Duality for Locally Compact Abelian Groups — Examples

Given: The group ${\mathbb{R}}^n$ with its standard topology and the dual identification constructed from the character classification of the line and the product-dual identification (Continuous characters of the real line are exponentials, Duals of finite products and of discrete direct sums).

Every continuous homomorphism $\phi :{\mathbb{R}}^n\to \mathbb{T}$ is $\phi (x)=e^{2\pi i\xi \cdot x}$ for a unique $\xi \in {\mathbb{R}}^n$, and $\xi \mapsto {\phi }_{\xi }$ is an isomorphism of topological groups ${\mathbb{R}}^n\to \widehat{{\mathbb{R}}^n}$: the classification of characters of the line supplies the algebraic bijection, and finite-product duality reduces the topology check to the line. On the line, compact $K$ is bounded, say $\mathrm{\mid }t\mathrm{\mid }\le M$; continuity of $s\mapsto e^{2\pi is}$ at $0$ makes $e^{2\pi i\xi t}$ uniformly close to $e^{2\pi i{\xi }_{0}t}$ on $K$ when $\mathrm{\mid }\xi -{\xi }_0\mathrm{\mid }$ is small. Conversely, if $\mathrm{\mid }\xi -{\xi }_0\mathrm{\mid }\ge ϵ$, then $t=1\mathrm{/}(2\mathrm{\mid }\xi -{\xi }_0\mathrm{\mid })\in [-1\mathrm{/}ϵ,1\mathrm{/}ϵ]$ gives $\mathrm{\mid }{e}^{2\pi i(\xi -{\xi }_0)t}-1\mathrm{\mid }=2$, so the uniform ball of radius $1$ on this compact interval forces $\mathrm{\mid }\xi -{\xi }_0\mathrm{\mid }<ϵ$. Uniform balls are compact-open neighbourhoods by The compact-open character group is a Hausdorff topological abelian group, and compactness and boundedness of these line sets follow from Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line. The value at $\pm \pi i$ is $-1$ by $\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$. At $n=0$ both groups are trivial. (Continuous characters of the real line are exponentials, Duals of finite products and of discrete direct sums, The Pontryagin dual with the compact-open topology)

$H^{\perp }=\{\gamma :\gamma (h)=1\text{ for every }h\in H\}$, and $e^{2\pi it}=1$ exactly when $t\in \mathbb{Z}$, while $e^{2\pi i(t+k)}=e^{2\pi it}$ for $k\in \mathbb{Z}$. (The annihilator of a subgroup, $\ker (\exp )=2\pi i\mathbb{Z}$, and $\exp z=\exp w$ exactly when $z-w\in 2\pi i\mathbb{Z}$, The integers as equivalence classes of pairs of naturals)

The quotient ${\mathbb{R}}^n\mathrm{/}H$ for $H={\mathbb{Z}}^k\times \{0{\}}^{n-k}$ is $(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^k\times {\mathbb{R}}^{n-k}$ with the product topology: each projection $\mathbb{R}\to \mathbb{R}\mathrm{/}\mathbb{Z}$ is continuous and open, since the saturation of an open set is the union of its integer translates. The finite product of these maps and identity maps is therefore continuous, open and surjective, with kernel $H$, and the induced quotient bijection is continuous and open. A continuous character on ${\mathbb{R}}^n$ constant on the cosets of $H$ factors through the quotient map, uniquely and continuously; conversely characters of the quotient pull back to characters of ${\mathbb{R}}^n$ that are trivial on $H$. (The quotient group $G\mathrm{/}N$ and coset product $(gN)(hN)=ghN$, The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, For a quotient map $q:X\to Y$, a map out of $Y$ is continuous iff its composite with $q$ is; a continuous map on $X$ constant on the fibres of $q$ factors uniquely through $q$; and a composite of quotient maps is a quotient map, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological)

For a real matrix $A$, a vector $\xi$ and $z\in {\mathbb{Z}}^n$ one has $(A^T\xi )\cdot z=\xi \cdot (Az)$ and $A^{-T}=(A^T{)}^{-1}$; the standard basis vectors $e_0,\dots ,e_{n-1}$ lie in ${\mathbb{Z}}^n$, and $y\in {\mathbb{R}}^n$ satisfies $y\cdot z\in \mathbb{Z}$ for all $z\in {\mathbb{Z}}^n$ exactly when $y\in {\mathbb{Z}}^n$. (Transpose is linear and involutive, and $(AB{)}^{\mathsf{T}}=B^{\mathsf{T}}{A}^{\mathsf{T}}$, The transpose $A^{\mathsf{T}}$ of a matrix, Invertible matrices and the general linear group ${\mathrm{GL}}_n(F)$, The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$, The integers as equivalence classes of pairs of naturals)

Given: The discrete group $\mathbb{Z}$ (The integers as equivalence classes of pairs of naturals, The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies) and the circle $\mathbb{T}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ (The multiplicative unit circle is a compact metrizable topological abelian group).

A homomorphism $\gamma :\mathbb{Z}\to \mathbb{T}$ is determined by $\gamma (1)$, and every $z\in \mathbb{T}$ occurs; since $\mathbb{Z}$ carries the discrete topology every such homomorphism is continuous. Hence $z\mapsto (n\mapsto z^n)$ is an algebraic isomorphism $\mathbb{T}\to \widehat{\mathbb{Z}}$. Compact subsets of the discrete space $\mathbb{Z}$ are finite (their cover by singletons has a finite subcover), so the preimage of each compact-open subbasic set is a finite intersection of open conditions on powers of $z$; the inverse is evaluation at $1$, whose preimages of open sets are subbasic open sets. Thus both directions are continuous. (The integers as equivalence classes of pairs of naturals, The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, The Pontryagin dual with the compact-open topology, Topological group: multiplication and inversion are continuous)

Every continuous homomorphism $\chi :\mathbb{T}\to \mathbb{T}$ is $\chi (z)=z^k$ for a unique $k\in \mathbb{Z}$. Indeed, writing $\mathbb{T}\cong \mathbb{R}\mathrm{/}\mathbb{Z}$ through the unit-circle isomorphism, $\chi$ corresponds to a continuous homomorphism $\psi :\mathbb{R}\to \mathbb{T}$ with $\psi (1)=1$; by the classification of characters of the line there is a unique $\xi \in \mathbb{R}$ with $\psi (t)=e^{2\pi i\xi t}$, and $\psi (1)=1$ forces $e^{2\pi i\xi }=1$, that is $\xi \in \mathbb{Z}$ by the kernel of the complex exponential. The dual of the compact circle is discrete by Compact groups have discrete duals and discrete groups have compact duals, so this bijection from the discrete group $\mathbb{Z}$ is a topological isomorphism. (The multiplicative unit circle is a compact metrizable topological abelian group, Continuous characters of the real line are exponentials, $\ker (\exp )=2\pi i\mathbb{Z}$, and $\exp z=\exp w$ exactly when $z-w\in 2\pi i\mathbb{Z}$, The Pontryagin dual with the compact-open topology)

For every locally compact Hausdorff abelian group $G$ the evaluation map ${\mathrm{\Phi }}_G(x)(\gamma ):=\gamma (x)$ is an isomorphism of topological groups $G\to \widehat{\widehat{G}}$. (Pontryagin biduality: the evaluation map is a topological isomorphism)

Given: The Axiom of Choice (The Axiom of Choice), the discrete additive group $\mathbb{Z}$ (The integers as equivalence classes of pairs of naturals, The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies) and the multiplicative circle $\mathbb{T}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ (The multiplicative unit circle is a compact metrizable topological abelian group).

The map $z\mapsto (n\mapsto z^n)$ is an isomorphism of topological groups $\mathbb{T}\to \widehat{\mathbb{Z}}$, because a homomorphism out of $\mathbb{Z}$ is determined by its value at $1$ and $\mathbb{Z}$ is discrete. Compact subsets of the discrete space $\mathbb{Z}$ are finite (their cover by singletons has a finite subcover), so the preimage of each compact-open subbasic set is a finite intersection of open conditions on powers of $z$; the inverse is evaluation at $1$, whose preimages of open sets are subbasic open sets. Thus both directions are continuous. Every continuous homomorphism $\chi :\mathbb{T}\to \mathbb{T}$ is $\chi (z)=z^k$ for a unique $k\in \mathbb{Z}$: writing $\mathbb{T}\cong \mathbb{R}\mathrm{/}\mathbb{Z}$, the character corresponds to a continuous homomorphism $\psi :\mathbb{R}\to \mathbb{T}$ with $\psi (1)=1$, the classification of characters of the line gives $\psi (t)=e^{2\pi i\xi t}$ for a unique real $\xi$, and $\psi (1)=1$ forces $\xi \in \mathbb{Z}$ by the kernel of the complex exponential. (The integers as equivalence classes of pairs of naturals, The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, The multiplicative unit circle is a compact metrizable topological abelian group, Continuous characters of the real line are exponentials, $\ker (\exp )=2\pi i\mathbb{Z}$, and $\exp z=\exp w$ exactly when $z-w\in 2\pi i\mathbb{Z}$, The Pontryagin dual with the compact-open topology)

Assuming the Axiom of Choice, $\mathbb{R}$ has a Hamel basis $B$ over $\mathbb{Q}$: every real is a finite $\mathbb{Q}$-linear combination of basis elements in exactly one way, a $\mathbb{Q}$-linear map $\mathbb{R}\to \mathbb{R}$ is determined by its values on $B$, and the coefficient of any basis element is a well-defined $\mathbb{Q}$-linear map. (Assuming the Axiom of Choice, $\mathbb{R}$ has a Hamel basis over $\mathbb{Q}$: there is $B\subseteq \mathbb{R}$ such that every real is a finite $\mathbb{Q}$-linear combination of elements of $B$ in exactly one way, and each basis vector carries a well-defined $\mathbb{Q}$-linear coefficient map)

A group homomorphism on a discrete group is continuous, and every function on a discrete space is continuous; composition of continuous homomorphisms is a continuous homomorphism, and $\mathbb{Z}\subseteq \mathbb{R}$ consists of the integer multiples of $1$. (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, Monoid homomorphism and group homomorphism, Topological group: multiplication and inversion are continuous, The integers as equivalence classes of pairs of naturals)

The assumed AC implies DC (AC implies DC implies countable choice), so biduality applies with its full choice hypotheses. With the compact-open topology the group $X=\widehat{\mathbb{Z}}$ is compact, because the dual of a discrete abelian group is compact under the Axiom of Choice, and Pontryagin biduality identifies its dual with $\mathbb{Z}$ through the evaluation isomorphism. (Compact groups have discrete duals and discrete groups have compact duals, Pontryagin biduality: the evaluation map is a topological isomorphism, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, The Pontryagin dual with the compact-open topology)