Normal bases for the space of continuous functions defined on a subset of \(\mathbb{Z}_ p\) (Q1896474)

Let \(K\) be a complete field extension of \(\mathbb{Q}_p\). Let \(a\), \(q\) be units of \(\mathbb{Z}_p\) such that \(q\) is not a root of unity. Let \(V_q\) be the closure in \(\mathbb{Q}_p\) of the set \(\{aq^n\mid n\geq 0\}\). It is shown that the Banach space \(C(V_q\to K)\) of continuous functions equipped with the uniform convergence (i.e. with the supremum norm) has an orthonormal basis \((\varepsilon_k)\) consisting of characteristic functions of suitably chosen discs. Moreover, necessary and sufficient conditions are given in order for the linear combinations of \(\varepsilon_k\) to form an orthonormal basis for \(C(V_q\to K)\).