unconditional convergence

Let $X$ be a topological vector space. Let $I$ be an index set and $x\_i\; \backslash in\; X$ for all $i\; \backslash in\; I.$

The series $\backslash textstyle\; \backslash sum\_\{i\; \backslash in\; I\}\; x\_i$ is called unconditionally convergent to $x\; \backslash in\; X,$ if

• the indexing set $I\_0\; :=\; \backslash left\backslash \{i\; \backslash in\; I\; :\; x\_i\; \backslash neq\; 0\backslash right\backslash \}$ is countable, and
• for every permutation (bijection) $\backslash sigma\; :\; I\_0\; \backslash to\; I\_0$ of $I\_0\; =\; \backslash left\backslash \{i\_k\backslash right\backslash \}\_\{k=1\}^\backslash infty$ the following relation holds: $\backslash sum\_\{k=1\}^\backslash infty\; x\_\{\backslash sigma\backslash left(i\_k\backslash right)\}\; =\; x.$

Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence $\backslash left(\backslash varepsilon\_n\backslash right)\_\{n=1\}^\backslash infty,$ with $\backslash varepsilon\_n\; \backslash in\; \backslash \{-1,\; +1\backslash \},$ the series

$$\backslash sum\_\{n=1\}^\backslash infty\; \backslash varepsilon\_n\; x\_n$$

converges.

If $X$ is a Banach space, every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general. Indeed, if $X$ 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 $X\; =\; \backslash R^n,$ by the Riemann series theorem, the series $\backslash sum\_n\; x\_n$ is unconditionally convergent if and only if it is absolutely convergent.

• {{annotated link|Absolute convergence}}
• {{annotated link|Modes of convergence (annotated index)}}
• {{annotated link|Absolute convergence#Rearrangements and unconditional convergence|Rearrangements and unconditional convergence/Dvoretzky–Rogers theorem}}
• {{annotated link|Riemann series theorem}}

{{reflist}}

• Ch. Heil: [http://www.math.gatech.edu/~heil/papers/bases.pdf A Basis Theory Primer]
• {{cite book

| last = Knopp

| first = Konrad

| title = Infinite Sequences and Series

| url = https://archive.org/details/infinitesequence0000knop

| url-access = registration

| isbn = 9780486601533

| publisher = Dover Publications

| year = 1956

}}

• {{cite book

| last = Knopp

| first = Konrad

| title = Theory and Application of Infinite Series

| publisher = Dover Publications

| year = 1990

| isbn = 9780486661650

}}

• {{cite book

| last=Wojtaszczyk

| first=P.

| title=Banach spaces for analysts

| year=1996

| publisher=Cambridge University Press

| isbn=9780521566759}}

{{Analysis in topological vector spaces}}

{{PlanetMath attribution|urlname=unconditionalconvergence|title=Unconditional convergence}}

Category:Convergence (mathematics)

Category:Mathematical analysis

Category:Series (mathematics)

Category:Summability theory