Let $X$ be a set. Consider $\mathbf{Q}_p$ and $\mathbf{Z}_p$ as the $p$-adic numbers and $p$-adic integers, respectively. For any finite subset $F \subseteq X$, one can construct the topological abelian group:
$$\prod_{x \in F} \mathbf{Q}_p \times \prod_{x \in X \setminus F} \mathbf{Z}_p$$
This group is equipped with the product topology. Define $\mathbf{Q}_p(X)$ as the union of these spaces, employing the colimit topology. Another description of $\mathbf{Q}_p(X)$ is as functions from $X$ to $\mathbf{Q}_p$, where almost all values reside in $\mathbf{Z}_p$.
The interesting aspect of this group is its nature as a locally compact Hausdorff abelian group that is self-dual, meaning that there is an isomorphism between the group and its Pontryagin dual:
$$\mathbf{Q}_p(X) \cong \mathrm{Hom}(\mathbf{Q}_p(X), S^1)$$
One can establish this fact utilizing Harmonic Analysis and properties of LCA groups. The construction of the inducing map for the isomorphism is as follows: There exists a natural pairing in $\mathbf{Q}_p(X)$ defined as:
$$\langle \xi, \eta \rangle = \sum_{x \in X} \xi(x)\eta(x) + \mathbf{Z}_p$$
Here, the values belong to $\mathbf{Q}_p/ \mathbf{Z}_p$ that we can consider it as a subset of $S^1$. Note that the above sum is finite as almost all elements are in $\mathbf{Z}_p$. This allows defining a map:
$$\mathbf{Q}_p(X) \to \mathrm{Hom}(\mathbf{Q}_p(X), S^1) \\ \xi \mapsto \langle \xi, - \rangle$$
Now, the question arises if a similar phenomenon occurs when substituting $\mathbf{Q}_p$ with $\mathbf{C}_p$ and $\mathbf{Z}_p$ with the closed unital ball of $\mathbf{C}_p$, denoted as $B$. Defining $\mathbf{C}_p(X)$ in a straightforward manner and using the same pairing as before, does this pairing induce an isomorphism of topological groups:
$$\mathbf{C}_p(X) \cong \mathrm{Hom}(\mathbf{C}_p(X), \mathbf{C}_p/B)$$
However, the problem is that $\mathbf{C}_p$ is not locally compact, therefore the old technique to prove the isomorphism will not work.
