Given a locally compact Abelian group $G$, there is a natural map $\Phi$ between it and its bidual $\hat{\hat{G}}$ given by $\Phi(x) = \phi_x$, where $\phi_x(\chi) = \chi(x)$.

The Pontryagin duality theorem states that $\Phi$ is a topological group isomorphism.

We will prove, without using the word ``representation", the Pontryagin duality theorem in 4 straightforward steps.