The question to follow has already been asked by the OP at https://math.stackexchange.com/questions/3454735/on-self-duality-of-non-archimedean-local-fields. Due to a lack of feedback, the OP felt compelled to post the same question here in hopes of a better feedback.

Let $K$ be a non-Archimedean local field. Its additive group $K^+$ is a locally compact Hausdorff abelian group. My question is the following:

Is $K^+$ isomorphic to its Pontryagin dual $\widehat{K^+}$ as a topological group?

**Remarks:**

1) Fix a non-trivial unitary character $\psi: K^+ \rightarrow \mathbb{C}^{\times}$. For each $a \in K$, the map $a\psi: K^+ \rightarrow \mathbb{C}^{\times}, \; x \mapsto \psi(ax)$ is a unitary character of $K^+$, and the map $a \mapsto a\psi$ gives an isomorphism of abstract groups from $K^+$ onto $\widehat{K^+}$ (see *e.g.* [1, sect. 1.17 Prop.]). Unfortunately, the argument given in [1, sect. 1.17 Prop.] does not prove (as far as I can see) that the previous map is an homeomorphism.

2) Let $p \in \mathbb{N}$ be the characteristic of the residue field of $K$. It is known that as a topological field, $K$ is isomorphic either to a finite extension of $\mathbb{Q}_p$ or to some field of formal Laurent series $\mathbb{F}_q((t))$ where $q$ is a power of $p$. Since the additive group of $\mathbb{Q}_p$ is known to be isomorphic (as a topological group) to its Pontryagin dual (see *e.g.* the introductory paragraphs in [2]), the question can be reduced to the case where $K=\mathbb{F}_q((t))$. But I can't find any references or arguments which hint to a positive answer.

Thank you in advance.

**References:**

[1] C. J. BUSHNELL AND G. HENNIART, *The Local Langlands Conjecture for GL(2)*, Springer, 2006.

[2] L. CORWIN, *Some remarks on self-dual locally compact abelian groups*, Trans. Amer. Math. Soc. **148** (1970), 613-622.