Duality in non-Archimedean vector spaces (Q2721293)

The paper is concerned with separation of convex sets and duality theory in non-archimedean (n.a.) functional analysis [see \textit{A. F. Monna}, ``Analyse Non-Archimédienne'', Berlin-Heidelberg (1970; Zbl 0203.11501)]. NEWLINENEWLINENEWLINEThe first three sections of the paper survey, in a short but very clear manner, some basic results on valued fields and n.a. topological vector spaces (TVS). NEWLINENEWLINENEWLINESection 4 is concerned with Hahn-Banach theorem and separation of convex sets in n.a. setting, meaning that for two closed subsets \(A,B\) of a n.a. TVS \(E\) there exists a continuous linear functional \(y\in E'\) such that \(\sup\{|y(a)|; a\in A\} < \inf \{|y(b)|: b\in B\}.\) Remark that there exists another notion of separation introduced by \textit{A. F. Monna}, Indag. Math. 26, 399-421 (1964; Zbl 0163.36103) [see also, \textit{S. Cobzaş}, Rev. Anal. Numer. Teor. Approx. 3, No. 1, 61-70 (1974)]. NEWLINENEWLINENEWLINEThe last section of the paper, Section 5, deals with duality theory for n.a TVS -- polars and the bipolar theorem, Mackey-Arens theorem.NEWLINENEWLINEFor the entire collection see [Zbl 0948.00027].