Density of characters of bounded \(p\)-adic analytic functions in the topological dual (Q2403578)

Let \(K\) be a field. A valuation on \(K\) is a function \(|\cdot|:K\to\mathbb{R}\) with {\parindent=0.8cm \begin{itemize}\item[(i)] \(|x|\geq 0\), \(x\in K\),\item[(ii)] \(|x|=0\Leftrightarrow x=0\),\item[(iii)] \(|x+y|\leq |x|+|y|\),\item[(iv)] \(|xy|= |x||y|\) for all \(x,y\in K\). \end{itemize}} A valuation \(|\cdot|\) defines a metric by \(d(x,y)= |x-y|\). \(K\) is called complete if it is complete relative to this metric. The metric \(d\) is called ultrametric if \(d(x,y)\leq \max\{d(x,z), d(z,y)\}\) for all \(x,y,z\in K\). An ultrametric field \((K,|\cdot|)\), i.e., a field in which the valuation \(|\cdot|\) satisfies \(|x+y|\leq \max\{|x|,|y|\}\) for all \(x,y\in K\), is called spherically complete if every decreasing sequence of balls in \(K\) has a nonempty intersection. A field \(K\) is said to be algebraically closed if each polynomial has a root in \(K\). The main results proved in this paper are decribed in the author's summary: ``(i) Let \(\mathbb{K}\) be an ultrametric complete algebraically closed field, let \(D\) be a disk \(\{x\in\mathbb{K} : |x|< R\}\) (with \(R\) in the set of absolute values of \(\mathbb{K}\)) and let \(A\) be the Banach algebra of bounded analytic functions in \(D\). The vector space generated by the set of characters of \(A\) is dense in the topological dual of \(A\) if and only if \(\mathbb{K}\) is not spherically complete. (ii) Let \(H(D)\) be the Banach algebra of analytic elements in \(D\). The vector space generated by the set of characters of \(H(D)\) is never dense in the topological dual of \(H(D)\).''