nLab restricted formal power series

Definition

Let $A$ be a commutative topological ring.

A formal power series

$$\sum c_{n_1,\ldots,n_p}X_1^{n_1}\cdots X_p^{n_p} \in A[[X_1,\ldots,X_p]]$$

is called restricted if for every neigbourhood $V$ of 0 in $A$, there is only a finite number of coefficients $c_{n_1,\ldots,n_p}$ not belonging to $V$. One can view this as the coefficients “converging to $0\in A$”.

If $A$ is linearly topologised (that is, it has a fundamental system of neighbourhoods of $0$ consisting of ideals) then restricted formal power series form a subring $A\{X_1,\ldots,X_p\} \subset A[[X_1,\ldots,X_p]]$.

Every derivative (in the formal sense) of a restricted formal power series is also restricted.

If $A$ is discrete, then a $A\{X_1,\ldots,X_p\} = A[X_1,\ldots,A_p]$, the ring of polynomials.