Summable family

A summable family is a term from functional analysis . It serves to generalize the concept of series for any families into a vector space .

Formal definition

Let be a normalized vector space. Be an index set and a family. Be . ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle I}$ ${\displaystyle (x_{i})_{i\in I}}$ ${\displaystyle {\hat {x}}\in X}$

The family is called summable to if and only if ${\displaystyle (x_{i})_{i\in I}}$ ${\displaystyle {\hat {x}}}$

${\displaystyle \forall \varepsilon >0\;\exists E_{0}\subseteq _{\text{endlich}}I\;\forall E_{0}\subseteq E\subseteq _{\text{endlich}}I:{\biggl \|}{\hat {x}}-\sum _{i\in E}x_{i}{\biggr \|}<\varepsilon }$

applies. So if for each a finite subset can be found in such a way that for all finite supersets that lie in, the sum in the norm deviates from less than . ${\displaystyle \varepsilon >0}$ ${\displaystyle E_{0}\subseteq I}$ ${\displaystyle E\supseteq E_{0}}$ ${\displaystyle I}$ ${\displaystyle \textstyle \sum _{i\in E}x_{i}}$ ${\displaystyle {\hat {x}}}$ ${\displaystyle \varepsilon }$

As with series, absolute summability can also be defined. The family is called absolutely summable to and only if is summable to one . ${\displaystyle (x_{i})_{i\in I}}$ ${\displaystyle {\hat {x}}}$ ${\displaystyle (\|x_{i}\|)_{i\in I}}$ ${\displaystyle s\in [0,\infty [}$

Ultimately, a family is called Cauchy-summable if and only if

${\displaystyle \forall \varepsilon >0\;\exists E_{0}\subseteq _{\text{endlich}}I\;\forall E\subseteq _{\text{endlich}}I\setminus E_{0}:{\biggl \|}\sum _{i\in E}x_{i}{\biggr \|}<\varepsilon }$

applies.

Remarks

• Absolute summability implies summability. The reverse is generally not true.
• If a family can be totaled, each subfamily can also be totaled. Summability is therefore a stronger criterion than simple convergence of series.
• Cauchy summability follows from summability. The reverse is true in Banach spaces . Cauchy summability is often easier to test.
• Be summable to and summable to and a scalar. Then applies . ${\displaystyle x}$ ${\displaystyle {\hat {x}}}$ ${\displaystyle y}$ ${\displaystyle {\hat {y}}}$ ${\displaystyle \lambda }$ ${\displaystyle \textstyle \sum _{i\in I}(x+\lambda y)={\hat {x}}+\lambda {\hat {y}}}$
• The carrier of a summable family is at most countable .

Individual evidence

1. ↑ ^{a b} Jürgen Heine: Topology and Functional Analysis . Oldenbourg, Munich 2011, ISBN 978-3-486-70530-0 , pp. 230 .