Unconditional convergence

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.

Definition

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} be a topological vector space. Let $I$ be an index set and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i \in X} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \in I.}

The series Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textstyle \sum_{i \in I} x_i} is called unconditionally convergent to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X,} if

• the indexing set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_0 := \left\{i \in I : x_i \neq 0\right\}} is countable, and
• for every permutation (bijection) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma : I_0 \to I_0} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_0 = \left\{i_k\right\}_{k=1}^\infty} the following relation holds: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\varepsilon_n\right)_{n=1}^\infty,} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon_n \in \{-1, +1\},} the series Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty \varepsilon_n x_n} converges.

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a Banach space, every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general. Indeed, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = \R^n,} by the Riemann series theorem, the series Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \sum_n x_n} is unconditionally convergent if and only if it is absolutely convergent.

See also

• Absolute convergence – Mode of convergence of an infinite series
• Modes of convergence (annotated index) – Annotated index of various modes of convergence
• Rearrangements and unconditional convergence/Dvoretzky–Rogers theorem – Mode of convergence of an infinite series
• Riemann series theorem – Unconditional series converge absolutely

References

• Ch. Heil: A Basis Theory Primer
• Knopp, Konrad (1956). Infinite Sequences and Series. Dover Publications. ISBN 9780486601533.
• Knopp, Konrad (1990). Theory and Application of Infinite Series. Dover Publications. ISBN 9780486661650.
• Wojtaszczyk, P. (1996). Banach spaces for analysts. Cambridge University Press. ISBN 9780521566759.

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