Unconditional convergence

Unconditional convergence is a topological property (convergence) related to an algebraic object (sum). It is an extension of the notion of convergence for series of countably many elements to series of arbitrarily many. It has been mostly studied in Banach spaces.

A series of numbers is unconditionally convergent if under all reorderings of the numbers, their sum converges to the same value as under the given ordering—their sum is not conditional on the particular arrangement. In contrast, a series is conditionally convergent if it converges but different orderings do not all converge to that same value.

Let ${\displaystyle X}$ be a topological vector space. Let ${\displaystyle I}$ be an index set and ${\displaystyle x_{i}\in X}$ for all ${\displaystyle i\in I}$.

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