Tate's proof that the character group of a local field is isomorphic to its additive group I'm reading Tate's thesis, and I'm stuck on one of his proofs. Let $k$ denote the completion of an algebraic number field at a prime divisor $p$. Tate claims that the additive group $k^+$ of $k$ is isomorphic, both topologically and algebraically, to its character group. Let $\xi \mapsto \chi(\xi)$ be a non-trivial character of $k^+$ and define a map from $k^+$ to $\widehat{k^+}$ by $\eta \mapsto \chi(\eta \xi)$. He proves that the map is injective and bicontinuous, but I don't understand why the map is surjective. In the proof he first shows that the image of the map is dense in $\widehat{k^+}$. By bicontinuity, the image is locally compact. Then he concludes, "Local compactness implies completeness and therefore closure..." (p. 309 of Cassels & Frohlich).
Could someone explain that sentence? How does local compactness imply completeness, and what does completeness even mean in the case of a character group? Is there a metric on the character group? I've looked in several books (e.g. Lang's Algebraic Number Theory, Ramakrishnan & Valenza) and could not find an explanation. I'd appreciate any help.

There is a proof of this fact in Bushnell, Henniart "The local Langlands conjecture for GL(2)", see 1.7, Proposition. In fact, it works for any nonarchimedean local field (not necessarily a $p$-adic field). Maybe the explanation there would help.
