Unconditional convergence

In mathematics, specifically functional analysis, a series is unconditionally convergent if all reorderings of the series converge. In contrast, a series is conditionally convergent if it converges but different orderings do not all converge to that same value. Unconditional convergence is equivalent to absolute convergence in finite dimensional vector spaces, but is a weaker property in infinite dimensions. In mathematics, specifically functional analysis, a series is unconditionally convergent if all reorderings of the series converge. In contrast, a series is conditionally convergent if it converges but different orderings do not all converge to that same value. Unconditional convergence is equivalent to absolute convergence in finite dimensional vector spaces, but is a weaker property in infinite dimensions. Let X {displaystyle X} be a topological vector space. Let I {displaystyle I} be an index set and x i ∈ X {displaystyle x_{i}in X} for all i ∈ I {displaystyle iin I} . The series ∑ i ∈ I x i {displaystyle extstyle sum _{iin I}x_{i}} is called unconditionally convergent to x ∈ X {displaystyle xin X} , if Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence ( ε n ) n = 1 ∞ {displaystyle (varepsilon _{n})_{n=1}^{infty }} , with ε n ∈ { − 1 , + 1 } {displaystyle varepsilon _{n}in {-1,+1}} , the series