A book that I'm currently reading (Fourier Analysis on Number Fields by Dinakar Ramakrishnana, Robert J. Valenza) claims that a continuous homomorphism $\mathbb{A}_k \to \widehat{\mathbb{A}_k}$ given by $y \to \psi_y(x)$ with $\psi_y(x) = \psi(yx)$ for some character $\psi \in \widehat{\mathbb{A}_k}$ is an isomorphism between an adele ring on a number field and its Pontryagin dual. However no proof is provided. I'm not sure where to start with a proof. I think you can show injectivity by showing that $\psi_y(x)=1$ iff $y=0$ but I have no idea how to show one-to-one correspondence. Any help would be greatly appreciated.
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(This should probably just be a comment, but I don't have enough reputation yet).
One can reduce this problem to proving that local fields are self-dual by theorem 5.4, which states that the Pontryagin dual of a restricted product is the restricted product of Pontryagin duals. The local case has been done here.
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