Let $L\subseteq\mathbb{C}_p$ be a finite extension of $\mathbb{Q}_p$, $r$ be a positive real number, and $f$ be a series $\sum_{n\in \mathbb{Z}} a_nz^n$ convergent in $D= \{x\in \mathbb{C}_p|0<v(x)\leq r \}$ where $a_n$ are elements in $L$. Then I want to know if the following are equivalent.

(1) $f$ is a bounded function in the metric of $\mathbb{C}_p$

(2) $f$ only has finitely many zeros in $D$

(3) the set $\{\lvert a_n\rvert\}$ is bounded as a subset of $\mathbb{R}$(in the Euclid metric)

Symbols: $\lvert a_n\rvert\mathrel{:=}p^{-v(a_n)}$ and $v$ is the valuation of $\mathbb{C}_p$ extended by the valuation on $\mathbb{Q}_p$.

Motivations: I want to use this to prove some properties of the Robba ring over $L$, e.g., $\varepsilon^\dagger$ is a field.

Thanks!

finitely manyzeros? $\endgroup$3more comments