Normal bases for the space of continuous functions defined on a subset of \(\mathbb{Z}_ p\). (English) Zbl 0840.46056

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 \(C(V_q\to K)\).


46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46E15 Banach spaces of continuous, differentiable or analytic functions
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