## A scalar field for which $$c_ 0$$ has no Hahn-Banach property.(English)Zbl 0830.46072

Summary: Let $$K$$ be a field with a valuation $$|\;|$$ of infinite rank for which $$(K,|\;|)$$ is complete, separable and such that the value group is countable. We show that the Banach space $$c_0$$ consisting of all null sequences in $$K$$ admits a closed subspace $$S$$ and a continuous linear function $$S\to K$$ that cannot be extended to a continuous linear function $$c_0\to K$$. This surprising result is in contrast not only to H. A. Keller’s example [Math. Z. 172, 41-49 (1980; Zbl 0423.46013)] of a separable inner product Banach space over $$K$$ resembling classical Hilbert space, but also to the Hahn-Banach property of $$c_0$$ over a field with a rank 1 valuation.

### MSC:

 46S10 Functional analysis over fields other than $$\mathbb{R}$$ or $$\mathbb{C}$$ or the quaternions; non-Archimedean functional analysis 46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators

### Keywords:

Hahn-Banach property; field with a rank 1 valuation

Zbl 0423.46013
Full Text:

### References:

 [1] Gross, H. and Künzi, U-M., On a class of orthomodular quadratic spaces. L’Enseignement Mathématique31 (1985), 187-212. · Zbl 0603.46030 [2] Keller, H.A., Ein nicht-klassischer Hilbertscher Raum. Math. Z.172 (1980), 41-49. · Zbl 0414.46018 [3] Ochsenius, H., Non-archimedean analysis when the value group has non-archimedean order. In : , Proceedings of the 2nd International Conference on p-adic Functional Analysis, N. De Grande-De Kimpe, S. Navarro and W.H. Schikhof, Editorial de la Universidad de Santiago de Chile (1994), 87-98. [4] van Rooij, A.C.M., Non-archimedean Functional Analysis. Marcel Dekker, New York (1978). · Zbl 0396.46061 [5] Warner, S., Topological fields. North Holland, Amsterdam (1989). · Zbl 0683.12014
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