Let $\mathbb{F_p}$ be the field with $p$-elements and $\mathbb{F_p}[[T]]$ denotes the power series ring of one variable.
Question: Can we say what will be the Pontryagin dual Hom$_{\text{cont}}$ $(\mathbb{F_p}[[T]], \mathbb{Q}_p/\mathbb{Z}_p)$ of $\mathbb{F_p}[[T]]$? where $\mathbb{Q}_p$ denotes the the field of p-adic numbers and $\mathbb{Z}_p$-is the p-adic integers.
All I can say that Hom$_{\text{cont}}$ $(\mathbb{F_p}[[T]], \mathbb{Q}_p/\mathbb{Z}_p)=$ Hom$_{\text{cont}} (\mathbb{F_p}[[T]], \mathbb{F}_p)$ is a co-free $\mathbb{F_p}[[T]]$-module. Since $\mathbb{F_p}[[T]]$ is a PID, co-free(injective modules) and divisible modules are the same. As a result Hom$_{\text{cont}}$ $(\mathbb{F_p}[[T]], \mathbb{Q}_p/\mathbb{Z}_p)$ becomes a $\mathbb{F_p}[[T]]$-divisible module, but I want to know the complete description of Hom$_{\text{cont}}$$(\mathbb{F_p}[[T]], \mathbb{Q}_p/\mathbb{Z}_p)$.
The Pontryagin dual sends limits to colimits so we get the filtered colimit of $\text{Hom}(\mathbb{F}_p[T]/T^n, \mathbb{F}_p)$. As a vector space this is $\mathbb{F}_p[S]$, and I believe the $\mathbb{F}_p[[T]]$-module structure is given by $T(S^i) = S^{i-1}$.