Unconditional convergence

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Order-independent convergence of a sequence

In mathematics, specifically functional analysis, a series is unconditionally convergent if all reorderings of the series converge to the same value. 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.

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Definition

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 i\in I.}

The series ∑ i ∈ I x i {\displaystyle \textstyle \sum _{i\in I}x_{i}} is called unconditionally convergent to x ∈ X , {\displaystyle x\in X,} if

the indexing set I 0 := { i ∈ I : x i ≠ 0 } {\displaystyle I_{0}:=\left\{i\in I:x_{i}\neq 0\right\}} is countable, and
for every permutation (bijection) σ : I 0 → I 0 {\displaystyle \sigma :I_{0}\to I_{0}} of I 0 = { i k } k = 1 ∞ {\displaystyle I_{0}=\left\{i_{k}\right\}_{k=1}^{\infty }} the following relation holds: ∑ k = 1 ∞ x σ ( i k ) = x . {\displaystyle \sum _{k=1}^{\infty }x_{\sigma \left(i_{k}\right)}=x.}

Alternative definition

Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence ( ε n ) n = 1 ∞ , {\displaystyle \left(\varepsilon _{n}\right)_{n=1}^{\infty },} with ε n ∈ { − 1 , + 1 } , {\displaystyle \varepsilon _{n}\in \{-1,+1\},} the series ∑ n = 1 ∞ ε n x n {\displaystyle \sum _{n=1}^{\infty }\varepsilon _{n}x_{n}} converges.

If X {\displaystyle X} is a Banach space, every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general. Indeed, if X {\displaystyle X} is an infinite-dimensional Banach space, then by Dvoretzky–Rogers theorem there always exists an unconditionally convergent series in this space that is not absolutely convergent. However, when X = R n , {\displaystyle X=\mathbb {R} ^{n},} by the Riemann series theorem, the series ∑ n x n {\textstyle \sum _{n}x_{n}} is unconditionally convergent if and only if it is absolutely convergent.

Definition

Alternative definition

See also