Unconditional convergence

In mathematical analysis, a series $\sum _{n=1}^{\infty }x_{n}$ in a Banach space X is unconditionally convergent if for every permutation $\sigma :\mathbb {N} \to \mathbb {N}$ the series $\sum _{n=1}^{\infty }x_{\sigma (n)}$ converges.

This notion is often defined in an equivalent way: A series is unconditionally convergent if for every sequence $(\varepsilon _{n})_{n=1}^{\infty }$, with $\varepsilon _{n}\in \{-1,+1\}$, the series

$\sum _{n=1}^{\infty }\varepsilon _{n}x_{n}$

converges.

Every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general. When X = Rⁿ, then, by the Riemann series theorem, the series $\sum x_{n}$ is unconditionally convergent if and only if it is absolutely convergent.

See also

• Modes of convergence (annotated index)

References

• Ch. Heil: A Basis Theory Primer
• K. Knopp: "Theory and application of infinite series"
• K. Knopp: "Infinite sequences and series"
• P. Wojtaszczyk: "Banach spaces for analysts"

Unconditional convergence at PlanetMath.