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Hahn-Banach theorem and duality theory on non-Archimedean locally convex spaces. (English) Zbl 1387.46052

Consider a field \(k\) with a non-trivial non-Archimedean (nA) valuation, with valuation ring \(O_k=\{\alpha\in k :|\alpha|\leq 1\}\) and residue field \(\bar k=O_k/U_k,\) where \(U_k=\{\alpha\in k :|\alpha|<1\}\) is the maximal ideal of \(O_k\). If \(k\) is locally compact, then one says that \(k\) is a local field. This is equivalent to the fact that \(k\) is complete, the valuation is discrete and the residue field is finite.
The author considers the extension problem for linear functionals on topological modules over \(k, O_k\) or \(\bar k\). Denoting by \(K\) one of these objects, one says that a pair \((M_0,M)\) of topological \(K\)-modules, where \(M_0\) is a submodule of \(M\), has the Hahn-Banach property (HBP) if every continuous \(K\)-linear homomorphism \(M_0 \to K\) admits a continuous \(K\)-linear extension \(M \to K\). One says that \(M\) has the HBP if \((M,M_0)\) has the HBP for any \(K\)-submodule \(M_0\) of \(M\). A. W. Ingleton [Proc. Camb. Philos. Soc. 48, 41–45 (1952; Zbl 0046.12001)] characterized the HBP in terms of spherical completeness (equivalent in the nA case to Nachbin’s binary intersection property). This implies that it holds for any seminormed \(K\)-vector space with \(K\in \{k, \bar k\}\). The author shows that HBP holds for any locally convex \(K\)-vector space for \(K=k\) (Theorem 3.1), and for \(K=\bar k\) (Theorem 3.8), thus giving partial extensions to Ingleton’s result.
The second section of the paper contains a study of linear topological modules, while the fourth one is concerned with the reflexivity of locally convex spaces over \(K\) (self-duality in the language of category theory).
The results are expressed in terms of category theory.

MSC:

46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
12J25 Non-Archimedean valued fields
16D90 Module categories in associative algebras

Citations:

Zbl 0046.12001
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