Let $T=\mathbb{R}/\mathbb{Z}$ denote the unit circle. For each $y\in\mathbb{R}^n$, define $\chi_y:\mathbb{R}^n\to T$ by $\chi_y(x) =xy+\mathbb{Z}$, where $xy$ denotes the scalar product of $x$ and $y$. Then $y\to \chi_y$ is a group isomorphism from $\mathbb{R}^n\to \hat{\mathbb{R}^n}$, where $\hat{\mathbb{R}^n}$ denotes Pontryagin dual of $\mathbb{R}^n$.
My question is that is this isomorphism also an isomorphism of topological groups, where Pontryagin dual has topology of compact convergence.