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.


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


Zbl 0423.46013
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