The property that a series still converges when the sequence
of its terms is arbitrarily rearranged. More exactly, a series
of elements of a linear space
in which the concept of a convergent sequence is defined is called
unconditionally convergent
if it converges after any rearrangement of its terms.
One approach to the study of unconditional convergence is the
study of unconditionally convergent series in metric vector (or topological) spaces,
[1]–[3].
Thus, for the unconditional convergence of the series
(*)
of elements of a Banach space
,
it is necessary and sufficient for each partial series
,
,
to be convergent
[4]
(the
Orlicz theorem).
Unconditional convergence of a numerical series is
equivalent to its absolute convergence (cf. the
Riemann theorem
on the rearrangement of the terms of a series). In general, if
is a finite-dimensional normed space, unconditional convergence
of a series is equivalent to convergence of the series
.
In an infinite-dimensional Banach space this is not valid.
Another direction of study concerns the properties of unconditionally
almost-everywhere convergent series of functions (or orthogonal series)
[5].
These properties are often quite different from the properties
of unconditionally convergent series in Banach spaces. For instance, the
analogue of the Orlicz theorem formulated above
is not valid for unconditional almost-everywhere convergence
[6].