# Unconditional summability

$$\sum_{n=1}^\infty a_n\label{*}\tag{*}$$
is called unconditionally summable by some summation method $A$ (unconditionally $A$-summable) if it is summable by this method to a sum $s$ whatever the ordering of its terms, where the value of $s$ may depend on the particular rearrangement (cf. Summation methods). The study of unconditional summability originated with W. Orlicz ; he showed, in particular, that if $\lim_{n\to\infty}a_n=0$, then absolute summability of the series by a linear regular method (cf. Regular summation methods) implies unconditional convergence. It was subsequently shown that this condition may be replaced by a weaker one: $\varliminf_{n\to\infty}a_n=0$ . Unconditional summability by a matrix method does not imply unconditional convergence; in fact, take the series $\sum_{n=1}^\infty1$. If $A$ is a regular matrix summation method and if the series \eqref{*} is unconditionally $A$-summable, then all its terms have the form $a_n=c+\eta_n$, where $c$ is a constant and the series with terms $\eta_n$ is absolutely convergent: $\sum_{n=1}^\infty|\eta_n|<\infty$; moreover, $c=0$ if the method $A$ does not sum the series $\sum_{n=1}^\infty1$ .
In the case of series of functions one distinguishes between summability in measure, everywhere summability, almost-everywhere summability, etc. For unconditional summability of a series of functions, the following statement is valid almost-everywhere: If the series $\sum_{n=1}^\infty f_n(x)$ of measurable functions $f_n$ on a set $E$ is unconditionally $A$-summable almost-everywhere on $E$, then the terms of this series have the form $f_n(x)=f(x)+\eta_n(x)$, where $f$ is a finite measurable function on $E$ and the series $\sum_{n=1}^\infty\eta_n(x)$ is unconditionally almost-everywhere convergent on $E$; also, $f=0$ if $A$ does not sum $\sum_{n=1}^\infty1$ .