Dual Group of $\Bbb{R}$

Using two theorem from the book "Pontryagin Duality and the Structure of Locally Compact Abelian Groups" wirtten by Sidney A. Morris the proof is as follows:

First the two theorems from the book:

Theorem 1 (Open Mapping Theorem)

Let G be a locally compact Group that is $\sigma$-compact; that is, $G = \displaystyle \bigcup_{n=1}^{\infty} A_n$ where each $A_n$ is a compact. Let $f$ be any continous homomorphism of $G$ onto a locally compact group $H$. Then $f$ is an open mapping, that means the image of every open set is open.

This theorem will help showing that the map is open and therefore, as N.H. hinted, an homeomorphism. Before we can apply this theorem we need to know if $ \hat{\mathbb{R}}$ is a locally compact group.

Theorem 2

Let G be an LCA-group (locall compact abelian) then $\hat{G}$ the character space, endowed with the compact open topology , is an LCA-Group.

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Claim: The map $$\Bbb{R} \to \hat{\Bbb{R}}, \quad y \mapsto \exp(ixy) $$ is an isomorphism and an homeomorphism.

Proof: $(\Bbb{R},+)$ endowed with the standard topology surely is an LCA-group. It follows by Theorem 2 that $\hat{\Bbb{R}}$ endowed with the compact open topology is LCA. (Here the compact open topology is equivalent to the topology induced by compact convergence.) Further it clearly is $\mathbb{R} = \displaystyle \bigcup_{n=1}^\infty [-n,n]$ and therefore it follows that $(\mathbb{R},+)$ is $\sigma$-compact. The given Map is a continous homomorphism and by applying Theorem 1 we follow that the given map is open. Therefore it is an homeomorphism.