Unconditional convergence

The property that a series still converges when the sequence of its terms is arbitrarily rearranged. More exactly, a series

$$ \tag{* } \sum _ { n= } 1 ^ \infty u _ {n} $$

of elements of a linear space $ E $ 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 $ E $, it is necessary and sufficient for each partial series $ \sum _ {k= 1 } ^ \infty u _ {n _ {k} } $, $ n _ {1} < n _ {2} < \dots $, 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 $ E $ is a finite-dimensional normed space, unconditional convergence of a series is equivalent to convergence of the series $ \sum _ {n= 1 } ^ \infty \| u _ {n} \| _ {E} $. 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].

References

[1]	S.S. Banach, "A course of functional analysis" , Kiev (1948) (In Ukrainian)
[2]	M.M. Day, "Normed linear spaces" , Springer (1958)
[3]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
[4]	W. Orlicz, "Beiträge zur Theorie der Orthogonalentwicklungen II" Studia Math. , 1 (1929) pp. 241–255
[5]	S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
[6]	P.L. Ul'yanov, "Divergent Fourier series" Russian Math. Surveys , 16 : 3 (1961) pp. 1–75 Uspekhi Mat. Nauk , 16 : 3 (1961) pp. 61–142