Let $A$ be a locally compact abelian (LCA) group. We shall denote its dual group by $\widehat{A}$. Pontryagin duality states there is a canonical isomorphism $\delta: A \to \widehat{\widehat{A}}$ of LCA groups between $A$ and its double dual, $\widehat{\widehat{A}}$. More precisely, $\delta: A \to \widehat{\widehat{A}}$ is given by $x\mapsto \delta_x$, where $\delta_x(\chi) = \chi(x)$ is the evaluation map.
Proposition $3.5.2$ in Principles of Harmonic Analysis by Anton Deitmar and Siegfried Echterhoff shows that the Pontryagin map $\delta$ is an (injective) continuous group homomorphism from $A$ to $\widehat{\widehat{A}}$.
I am wondering if the Pontryagin map $\delta$ is also uniformly continuous. For topological groups, uniform continuity is defined in the following sense:
For topological groups $G$ and $H$, a map $f: G\to H$ is left uniformly continuous if for every open neighborhood $V$ of the identity in $H$ there exists an open neighborhood $U$ of the identity in $G$ such that $f(x)^{-1} f(y) \in V$ for all $x,y\in G$ satisfying $x^{-1}y\in U$.
Since we are working with abelian groups, left and right uniform continuity are equivalent notions.
It seems reasonable to expect uniform continuity since for $A = \Bbb Z$ or $\Bbb T$, the Pontryagin map turns out to be the identity map under the identifications $\widehat{\Bbb Z} \cong \Bbb T$ and $\widehat{\Bbb T} \cong \Bbb Z$.
Thanks for any help!
Every continuous group homomorphisms between topological groups is (left) uniformly continuous: If $V$ is a neighborhood of the identity in $H$, then by continuity there exists a neighborhood $U$ of the identity in $G$ such that $f(U)\subset V$. If $g^{-1}h\in U$, then $f(g)^{-1}f(h)=f(g^{-1}h)\in f(U)\subset V$.