Normal bases for the space of continuous functions defined on a subset of $\mathbb{Z}_p$

Abstract

Let $K$ be a non-archimedean valued field which contains $\Bbb Q_p$ and suppose that $K$ is complete for the valuation $|\cdot|$, which extends the $p$-adic valuation. $V_q$ is the closure of the set $\{aq^n|n=0,1,2,\dots\}$ where $a$ and $q$ are two units of $\Bbb Z_p$, $q$ not a root of unity. $C(V_q\rightarrow K)$ is the Banach space of continuous functions from $V_q$ to $K$, equipped with the supremum norm. Our aim is to find normal bases $(r_n(x))$ for $C(V_q\rightarrow K)$, where $r_n(x)$ does not have to be a polynomial.