An approximation theorem for \(p\)-adic linear forms (Q2751740)

Let \(K\) be a complete non-Archimedean non-trivially valued field with valuation \(|\cdot|: K\to [0,\infty)\). \(K\) is called spherically complete if each nested sequence of balls has a non-empty intersection. A subset \(A\) of a \(K\)-vector space \(E\) is called absolutely convex if it is a module over the ring \(\{\lambda\in K:|\lambda|\leq 1\}\). Let \(E^*\) be the algebraic dual of \(E\). For \(S\subset E\) let \(S^0= \{f\in E^*: |f(x)|\leq 1\) for all \(x\in S\}\), for \(T\in E^*\) let \(T^-= \{x\in E:|f(x)|\leq 1\) for all \(f\in T\}\). \(S\) is called polar if \(S^{00}= S\). A seminorm \(p\) on \(E\) is called polar if \(p= \sup\{|f|: f\in E^*,(f)\leq p\}\). A locally convex space \((E,\tau)\) is called polar if \(\tau\) is generated by some family of polar seminorms or, equivalently, if there exists a neighbourhood based of \(0\) consisting of polar sets. A seminorm \(p\) on a \(K\)-vector space \(E\) is of countable type if there exists a countable set \(X\subset E\) such that the linear \(\text{span}[X]\) of \(X\) is dense in \(E\) with respect to the topology induced by \(\{p\}\). A locally convex space is of countable type if each continuous seminorm is of countable type. In this paper the following \(p\)-adic version of the approximation theorem is proved.NEWLINENEWLINENEWLINEApproximation theorem for linear forms: Let \((E,\tau)\) be a locally convex space over \(K\), let \(A\in E\) be absolutely convex. If \(K\) is not spherically complete assume that \((E,\tau)\) is of countable type. Let \(f\in E^*\) be such that \(f|A\) is continuous. Then, for each \(\varepsilon> 0\) there is a \(g\in E^*\) such that \(|g-f|\leq \varepsilon\) on \(A\).NEWLINENEWLINENEWLINEApplying this theorem an alternative proof of the non-Archimedean Grothendieck completeness theorem; a first proof was given by \textit{A. K. Katsaras} [Bull. Inst. Math. Acad. Sin. 19, No. 4, 351-354 (1991; Zbl 0780.46043)] is also given in this paper.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00058].