The van der Put base for \(C^ n\)-functions (Q1326945)

Let \(K\) be an algebraic extension of \(Q_p\), the field of \(p\)-adic numbers, and \(Z_p\) the ring of \(p\)-adic integers. Write \(\varphi_1 f(x_1, x_2)= (f(x_2)- f(x_1))/ (x_2- x_1)\) and \(\varphi_2 f(x_1, x_2, x_3)= (\varphi_1 f(x_2, x_3)- \varphi_1 f(x_1, x_3))/ (x_2- x_1)\) where \(x_1\), \(x_2\), \(x_3\) are all distinct. Then \(C^2 (Z_p\to K)\) denotes the Banach space of \(f: Z_p\to K\) such that \(\varphi_2 f\) can be extended to a continuous function on \(Z^3_p\). The author constructed an orthonormal base for \(C^2 (Z_p\to K)\) and claimed that the same holds for \(C^n (Z_p\to K)\). The proof is highly computational. For reference, see \textit{W. H. Schikhof} [Ultrametric calculus: An introduction to \(p\)-adic analysis (1984; Zbl 0553.26006)].