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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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