The Hahn-Banach extension property in a class of normed spaces. (English) Zbl 0603.46070

Normed spaces over non-archimedean valued fields whose norm does not verify the strong triangle inequality are called A-normed spaces in the literature. In this paper we study several properties of this kind of spaces related, most of them, to extension properties of bounded linear functionals. In particular, we prove that, under certain conditions, there exists no A-normed space verifying either the t-bounded extension property \((0<t\leq 1)\) or the Hahn-Banach extension property.


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
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