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While trying to understand a certain topological ring better, I stumbled onto the following question.

Suppose $I$ is a fixed infinite index set, $R$ is a topological ring and $(x_i)_{i\in I}$ is a family of elements of $R$ such that for all families $(a_i)_{i\in I}$ of elements of $R$ the series $\sum_{i\in I} a_i x_i$ converges, i.e. the net $E\mapsto\sum_{i\in E} a_i x_i$ converges where $E$ ranges over finite subsets of $I$.

Under what conditions on $R$ can we conclude that $x_i = 0$ for almost every $i$ ? I know some examples for this, but I'd like to have a general criterion if it exists.

Examples:

  • If $R$ is discrete, it works.
  • If $R$ is any topological field except those with the indiscrete topology, it works.
  • If $R$ is a normed algebra, it works.

Counterexamples:

  • $R$ indiscrete.
  • $R=\mathbb{Z}_p$, $I=\mathbb{N}$, $x_i=p^i$ or more generally: $R$ any DVR with maximal ideal $\mathfrak{m}$, $I=\mathbb{N}$, $x_i\in\mathfrak{m}^i$.
  • Even more general: $R$ a first countable topological ring with a neighborhood base of zero$\mathcal{U}=\{U_0\supseteq U_1\supseteq\ldots\}$ consisting of nonzero ideals, $I:=\mathbb{N}$, and $x_i\in U_i$. This includes all valuation rings $R$ with archimedean value group (i.e. $\Gamma=Quot(R)^\times / R^\times \leq \mathbb{R}$ as ordered groups) with their natural topology (i.e. $U_\gamma:=\{x\in R | \nu(x)>\gamma\}$ is a neighborhood base)
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additional examples: $\:$ any topological division ring except those with the indiscrete topology $\hspace{1.11 in}$ –  Ricky Demer Oct 18 '13 at 16:03
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