Banach-Alaoglu Theorem over spherically complete non-Archimedean fields

In the general non-Archimedean case, instead of using the concept of compactness, it is more suitable to use the concept of compactoidness. I think the theorem you are looking for is enter image description here

(page 273) all the details and preliminaries are in the spectacular book: Locally Convex Spaces over non-Arquimedean Valued Fields, Cambridge University Press - [C.Perez-Garcia, W.H.Schikhof] - 2010


Chilote has pointed to the right notion for the general case. I will answer the literal question.

The Banach-Alaoglu theorem (using the usual topological notion of compactness) cannot hold for normed spaces over a field with valuation $(k,|\cdot|)$ if the unit ball of $k$ is noncompact. The reason is that the weak-* dual of $k$ is isomorphic to $k$ with its original topology.

The spherical completion of the algebraic closure of $\mathbb{Q}_p$, for $p$ a prime, is a spherically complete field whose unit ball is noncompact.

Tags:

Banach Spaces

Hahn Banach Theorem

Functional Analysis

P Adic Number Theory

Nonarchimedian Analysis

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