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Does anyone know if the Hahn-Banach theorem is true for every locally compact field? Specifically, let $F$ be a finite algebraic extension of either $Q_p$, the $p$-adic completion of $Q$, or of $S_p$, the $t$-adic completion of $F_p\{t\}$, the field of formal Laurent series over the field of $p$ elements. Then is $F$ injective with respect to topological embeddings in the category of normed linear $F$-spaces?

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There exists a $p$-adic Hahn Banach theorem.

Reference Alain M. Robert. A course in p-adic analysis. Graduate text in Math.

Theorem (Ingleton)

Let $K$ be a spherically complete ultrametric field. $E$ a $K$-normed space and $V$ a subspace of $E$. For every bounded linear functional $\phi$ defined on $V$, there exists a bounded linear functional $\psi$ defined on $E$ whose restriction to $V$ is $\phi$ and such that $\|\phi\|=\|\psi\|$.

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    $\begingroup$ For those readers who do not have immediate access to that text, could you please state the theorem as given in the text? $\endgroup$ – Todd Trimble Feb 11 '16 at 14:11
  • $\begingroup$ I know what a complete ultrametric field is (and my fields are), but i have never seen the phrase "spherically complete" before. If my fields are that, then this answers my question completely. Never mind, I just googled it and, since the balls are compact, the result follows. Thanks. $\endgroup$ – Michael Barr Feb 11 '16 at 15:44
  • $\begingroup$ en.wikipedia.org/wiki/Spherically_complete_field $\endgroup$ – Tsemo Aristide Feb 11 '16 at 15:45

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