Pontryagin biduality: the evaluation map is a topological isomorphism

Given: A locally compact Hausdorff abelian group $G$ with dual $\widehat{G}$ and bidual $\widehat{\widehat{G}}$, and the evaluation map $\mathrm{\Phi }$.

$\mathrm{\Phi }$ is a continuous group homomorphism whose image is a subgroup of $\widehat{\widehat{G}}$; the sets $N_{x_1}(K,ϵ)=\{x:\mathrm{\mid }\chi (x)-\chi (x_1)\mathrm{\mid }<ϵ\text{ for all }\chi \in K\}$ over compact $K\subseteq \widehat{G}$ and $ϵ>0$ form a neighbourhood basis at $x_1$, and $\mathrm{\Phi }$ is a homeomorphism onto its image. (Compact-open neighbourhoods on the dual give a neighbourhood basis on the group, The Pontryagin dual with the compact-open topology, Continuity of a map of topological spaces at a point and globally)

$\mathrm{\Phi }$ is injective: continuous characters separate points. (Continuous characters separate points of an LCA group)

The dual of a locally compact Hausdorff abelian group is again locally compact Hausdorff and abelian, so both $\widehat{G}$ and $\widehat{\widehat{G}}$ are locally compact Hausdorff abelian, and every point of $\widehat{\widehat{G}}$ has a neighbourhood basis of compact sets. (The dual of a locally compact abelian group is locally compact abelian, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not)

A subgroup of a Hausdorff topological group which is locally compact in the subspace topology is closed. (A locally compact subgroup of a Hausdorff topological group is closed, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, Topological group: multiplication and inversion are continuous)

Bump on the dual of $\widehat{G}$. Applied to the locally compact abelian group $\widehat{G}$ with its Haar measure $m_{\widehat{G}}$: for ${\xi }_0\in \widehat{\widehat{G}}$ and a compact neighbourhood $K$ of ${\xi }_0$ there is $f\in L^1(\widehat{G},m_{\widehat{G}})$ with $\widehat{f}\ge 0$ on $\widehat{\widehat{G}}$, $\widehat{f}({\xi }_0)>0$ and $\widehat{f}=0$ on $\widehat{\widehat{G}}\setminus K$, where $\widehat{f}(\xi )={\int }_{\widehat{G}}f(\gamma )\overline{\xi (\gamma )}d{m}_{\widehat{G}}(\gamma )$. (Compactly supported nonnegative transform bumps on the dual, The Fourier transform on an LCA group, A compact identity neighbourhood in the dual)

A finite regular complex Borel measure $\mu$ on $\widehat{G}$ whose inverse transform $x\mapsto {\int }_{\widehat{G}}\gamma (x)d\mu (\gamma )$ vanishes for every $x\in G$ is zero. (Fourier-Stieltjes transforms determine finite Radon measures, Regular complex Borel measures)

For $f\in L^1(\widehat{G})$ the measure $\mu =f{m}_{\widehat{G}}$ is finite and regular: approximate $f$ in $L^1$ by $h_n\in C_c(\widehat{G})$, each $h_n{m}_{\widehat{G}}$ is finite regular as follows. For bounded $h$ supported in compact $K$, outer Haar approximations $O_1\supseteq E\cap K$ and $O_2\supseteq K\setminus E$ give an open superset $O_1\cup (\widehat{G}\setminus K)$ of $E$ and a compact subset $K\setminus O_2$ of $E$, with weighted errors bounded by $\mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}$ times the arbitrarily small Haar errors (In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones, A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact). Finally, $\mathrm{\parallel }(f-h_n)m_{\widehat{G}}{\mathrm{\parallel }}_{TV}=\mathrm{\parallel }f-h_n{\mathrm{\parallel }}_1\to 0$, while total-variation limits of finite regular complex measures are finite regular (outer and inner regularity transfer from an approximant with error control). (A complex L^1 density defines a complex measure whose total variation is |h| dmu, Radon measure on an LCH space, C_c(X) is dense in L^p(mu) for a Radon measure, Compact support, $C_c(X)$, and $C_0(X)$)