Tensor products of topological abelian groups and Pontryagin duality

Let $G$ be the group of all $\text{ZZ}$-valued homomorphisms of the Baer-Specker group ${\text{ZZ}}^{\text{NN}}$. The group $G$ is algebraically isomorphic to ${\text{ZZ}}^{(\text{NN})}$, the infinite direct sum of the group of integers, and equipped with the topology of pointwise convergence on ${\text{ZZ}}^{\text{NN}}$, becomes a non reflexive prodiscrete group. It was an open question to find its dual group $\widehat{G}$. Here, we answer this question by proving that $\widehat{G}$ is topologically isomorphic to ${\text{ZZ}}^{\text{NN}}{\otimes }_{\mathcal{Q}}\text{TT}$, the (locally quasi-convex) tensor product of ${\text{ZZ}}^{\text{NN}}$ and $\text{TT}$. Furthermore, we investigate the reflexivity properties of the groups of $C_p(X,\text{ZZ})$, the group of all $\text{ZZ}$-valued continuous functions on $X$ equipped with the pointwise convergence topology, and $A_p(X)$, the free abelian group on a $0$-dimensional space $X$ equipped with the topology $t_p(C(X,\text{ZZ}))$ of pointwise convergence topology on $C(X,\text{ZZ})$. In particular, we prove that $\widehat{A_p(X)}\simeq C_p(X,\text{ZZ}){\otimes }_{\mathcal{Q}}\text{TT}$ and we establish the existence of $0$-dimensional spaces $X$ such that $C_p(X,\text{ZZ})$ is Pontryagin reflexive.