## Normal bases for non-archimedean spaces of continuous functions.(English)Zbl 0804.46088

Summary: $$K$$ is a complete non-archimedean valued field and $$M$$ is a compact, infinite, subset of $$K$$. $$C(M\to K)$$ is the Banach space of continuous functions from $$M$$ to $$K$$, equipped with the supremum norm. Let $$(p_ n(x))$$ be a sequence of polynomials, with $$\deg p_ n= n$$. We give necessary and sufficient conditions for $$(p_ n(x))$$ to be a normal basis for $$C(M\to K)$$. In the rest of the paper, $$K$$ contains $$\mathbb{Q}_ p$$, and $$V_ q$$ is the closure of the set $$\{aq^ n\mid n=0,1,2\dots\}$$, where $$a$$ and $$q$$ are two units of $$\mathbb{Z}_ p$$, $$q$$ not a root of unity. We give necessary and sufficient conditions for a sequence of polynomials $$(r_ n(x))$$ $$(\deg r_ n= n)$$ to be a normal basis for $$C(V_ q\to K)$$. Furthermore, if we define ${x\brace 0}= 1,\;{x\brace n}= {(x/a-1)(x/(aq)- 1)\cdots (x/(aq^{n-1})- 1)\over (q^ n- 1)\cdots (q- 1)}\quad\text{if } n\geq 1,$ and if $$(j_ n)$$ is a sequence in $$\mathbb{N}_ 0$$, then we show that the sequence of polynomials $$\Bigl({x\brace n}^{j_ n}\Bigr)$$ forms a normal 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 46E15 Banach spaces of continuous, differentiable or analytic functions 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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