ScalingStacks

Definition 2.38. For a multi-radius $\boldsymbol{r}=(r_{1},\dots,r_{n})\in\mathbb{R}^{n}$, the algebra

$$k\{r_{1}^{-1}T_{1},\dots,r_{n}^{-1}T_{n}\}=\{f=\sum_{J\in\mathbb{N}^{n}}^{\infty}a_{J}\boldsymbol{T}^{J}:a_{J}\in k,\lvert a_{J}\rvert\boldsymbol{r}^{J}\to 0\text{ as }|J|\to\infty\}$$

is called the Tate algebra over $k$ with multi-radius $\boldsymbol{r}$. Denote it by $\mathcal{T}_{n}(\boldsymbol{r})$. It is a $k$-Banach algebra with respect to the Gauss norm of multi-radius $\boldsymbol{r}$ defined by

$$\vvvert f\vvvert_{\mathcal{T}_{n}(\boldsymbol{r})}=\max_{J}\lvert a_{J}\rvert\boldsymbol{r}^{J}$$

One can define Tate algebra over other complete ultra-metric valued fields.