3 Gauss Norm
Thus far we have shown restricted power series over a non-archimedean normed ring \(R\) for a parameter \(c\) forms a ring. Let us denote this ring by \(\mathcal{F}\). Next, let us consider the following function \(| \cdot |_c : \mathcal{F} \to \mathbb {R}\) with \(f = \sum a_n x^n\)
\[ |f|_c = \sup _{i \in \mathbb {N}} |a_n|c^n \]
The pair \((\mathcal{F}, | \cdot |_c)\) is a non-archimedean normed ring.
Moreover, if we restrict ourselves to polynomials we can strenghten this.
The pair \((R[x], | \cdot |_c)\) is a non-archimedean valued ring. That is, \(| \cdot |_c\) is an absolute value.