Absolutely convergent series

The unconditional convergent series is a term from functional analysis that describes a certain convergence behavior of series . One speaks of unconditional convergence of a series if the convergence is insensitive to rearrangements of the series . In the finite-dimensional this is equivalent to absolute convergence, in the infinite-dimensional it is no longer the case.

definition

Let be a topological vector space . Be an index set and for all . A series is said to necessarily converge to if ${\displaystyle X}$ ${\displaystyle I}$ ${\displaystyle x_{i}\in X}$ ${\displaystyle i\in I}$
${\displaystyle \textstyle \sum _{i\in I}x_{i}}$ ${\displaystyle x\in X}$

• the index set is countable and ${\displaystyle I_{0}:=\{i\in I:x_{i}\neq 0\}}$
• for every bijective mapping the equation ${\displaystyle a\colon \mathbb {N} \rightarrow I_{0}}$

${\displaystyle \lim _{n\to \infty }\sum _{j=1}^{n}x_{a(j)}=x}$

applies.

This term is mostly examined in Banach spaces , but can also be considered in normalized , locally convex or, as above, generally in topological vector spaces .

Applications

• With the help of this definition z. For example, in a topological vector space, introduce the usual concept of a "convergent sum of subspaces" as an extension of the already known sum of subspaces:
  • Sum of subspaces:
    ${\displaystyle \sum _{i\in I}U_{i}:=\left\{\sum _{i\in I}u_{i}:u_{i}\in U_{i},u_{i}=0{\text{ für fast alle }}i\in I\right\}}$
  • Extension "Convergent sum of subspaces": It is particularly important here that the value of the series does not depend on the rearrangement. Otherwise the elements would not be well defined.
    ${\displaystyle \sum _{i\in I}U_{i}:=\left\{\sum _{i\in I}u_{i}:u_{i}\in U_{i},\sum _{i\in I}u_{i}{\text{unbedingt konvergent}}\right\}}$
    
• The Birkhoff integral for Banach space valued functions is defined with the help of the unconditional convergence in Banach spaces.

Relation to absolute convergence

Riemann's theorem

→ Main articles : Riemann rearrangement theorem  and Steinitz rearrangement theorem

Let be the underlying Banach space and a countable index set. Then one of Riemann's theorem says that the series converges unconditionally if and only if it converges absolutely . ${\displaystyle X:=\mathbb {R} ^{n}}$ ${\displaystyle I}$ ${\displaystyle \textstyle \sum _{i\in I}x_{i}}$

Dvoretzky-Rogers theorem

→ Main article : Dvoretzky-Rogers theorem

In infinite-dimensional spaces the unconditional convergence and the absolute convergence are no longer equivalent. This is what the Dvoretzky-Rogers theorem, named after Aryeh Dvoretzky and Claude Ambrose Rogers , says . Precisely it says that in every infinite-dimensional Banach space there exists an unconditionally convergent series that does not absolutely converge. The inversion, according to which every absolutely convergent series necessarily converges, also holds in the infinite-dimensional case.

See also

• Weak unconditional Cauchy series

literature

• Dirk Werner : Functional Analysis . 6th, corrected edition. Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 , p. 232 ff.

Web links

• Chi Woo et al. a .: Unconditional convergence . In: PlanetMath . (English)
• BI Golubov: Unconditional convergence . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).