Prove that the Pontryagin dual of $\mathbb{R}$ is $\mathbb{R}$.

Question

From Wikipedia: the group of real numbers $\mathbb{R}$, is isomorphic to its own dual; the characters on $\mathbb{R}$ are of the form $r \to e^{i\theta r}$.

How can I prove this assertion?

Answer 1

This assertion is shown in the appendix on page $10$ of K. Conrad's notes about the character group of $\mathbb{Q}$. His proof for $\mathbb{R}$ is elementary, not using integrals (which is done in Conway's book on functional analysis). For more references see also this MSE question.