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

### MSC:

 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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