Rearranging rows

The rearrangement of series is studied in mathematics when studying the convergence of infinite sums called series . It is about the question of which limit values ​​of the series can be changed by rearranging the summands, i.e. H. by changing their order. In the case of real series, the Riemann rearrangement theorem provides information about the possible series sums; the situation in finite-dimensional vector spaces is treated exhaustively in Steinitz's rearrangement theorem .

Many statements about convergent series in finite-dimensional spaces lose their validity in infinite-dimensional spaces. Generalizations of Steinitz's rearrangement theorem can only be obtained under additional conditions. The focus of this article is the rearrangement of rows in infinite-dimensional spaces. Therefore, in contrast to Riemann's and Steinitz's rearrangement theorems, which can be assigned to classical analysis , functional analytical methods and conceptualizations play an important role.

Convergence terms

It is a Banach space . ${\displaystyle (X,\|\cdot \|)}$

• A series in is called convergent if the sequence of the partial sums converges. ${\displaystyle \textstyle \sum _{n=1}^{\infty }x_{n}}$ ${\displaystyle X}$ ${\displaystyle \textstyle \left(\sum _{n=1}^{N}x_{n}\right)_{N}}$
• The series is called unconditionally convergent if it converges for every permutation , that is, if every rearrangement of the series converges. ${\displaystyle \textstyle \sum _{n=1}^{\infty }x_{\sigma (n)}}$ ${\displaystyle \sigma \in S({\mathbb {N} })}$
• One speaks of perfect convergence if converges for each choice , that is, the series converges for each choice of signs of the summands. ${\displaystyle \textstyle \sum _{n=1}^{\infty }\alpha _{n}x_{n}}$ ${\displaystyle \alpha _{n}\in \{-1,1\}}$
• ${\displaystyle \textstyle \sum _{n=1}^{\infty }x_{n}}$is called sub- series convergent if it converges for every increasing sequence . ${\displaystyle \textstyle \sum _{k=1}^{\infty }x_{n_{k}}}$ ${\displaystyle n_{1}<n_{2}<n_{3}<\ldots }$
• The series is called absolutely convergent if . ${\displaystyle \textstyle \sum _{n=1}^{\infty }\|x_{n}\|<\infty }$

Problem

For an episode in be ${\displaystyle (x_{n})_{n}}$ ${\displaystyle X}$

${\displaystyle \,SM((x_{n})_{n}):=\left\{s\in X;\,\exists \sigma \in S({\mathbb {N} }):\,s=\sum _{n=1}^{\infty }x_{\sigma (n)}\right\}}$

the amount of all sums that can be obtained by rearranging the series , in short the sum amount of the sequence. The question arises, what can be said about the structure of this set. ${\displaystyle \textstyle \sum _{n=1}^{\infty }x_{n}}$

The finite-dimensional case

→ Main article : Steinitz's rearrangement theorem

The finite-dimensional case is treated exhaustively by Steinitz's rearrangement theorem. For a sequence is the subspace of the so-called convergence Functional . If the series is convergent, then where is the set all for which holds for all . In particular, there is always an affine subspace . ${\displaystyle (x_{n})_{n}}$ ${\displaystyle \textstyle \Gamma ((x_{n})_{n}):=\{f\in X':\sum _{n=1}^{\infty }|f(x_{n})|<\infty \}}$ ${\displaystyle \Sigma x_{n}}$ ${\displaystyle \textstyle SM((x_{n})_{n})=\sum _{n=1}^{\infty }x_{n}+(\Gamma (x_{n})_{n})^{0}}$ ${\displaystyle (\Gamma (x_{n})_{n})^{0}}$ ${\displaystyle x\in X}$ ${\displaystyle f(x)=0}$ ${\displaystyle f\in \Gamma ((x_{n})_{n})}$ ${\displaystyle SM((x_{n})_{n})}$

Furthermore, the following statements are equivalent for a series in a finite-dimensional space: ${\displaystyle \textstyle \sum _{n=1}^{\infty }x_{n}}$

• The series converges absolutely.
• The series necessarily converges.
• The series converges perfectly.
• The series is sub-series convergent.
• ${\displaystyle SM(x_{n})_{n}}$ is one element.

With regard to the above problem, the question arises whether these statements are also valid in infinite-dimensional spaces.

The problem 106

The question of the structure of the sums in infinite-dimensional spaces was first asked in 1935 by Stefan Banach as Problem 106 in the so-called Scottish Book . This is a notebook kept in the Scottish Café in Lemberg , in which the Lviv functional analysts and their guests recorded mathematical problems. Stefan Banach entered the assumption that the total amount was always affine and promised a bottle of wine to clarify the question, such prices were quite common for the problems posed here. In the Scottish book there is already a counterexample to this assumption, without an author being given, Józef Marcinkiewicz is considered the likely author after a manuscript analysis.

With this counterexample it was clear that a statement analogous to Steinitz's rearrangement theorem does not apply in the infinite-dimensional case. The examples known at the time were constructed in such a way that the sum set was still a subgroup of the additive group of the Banach space shifted by a constant vector . It was not until 1989 that MI Kadets and Krzysztof Wozniakowski, and independently PA Kornilow, were able to give examples of series for which the sum amount is not a shifted subgroup. It turned out that in every infinite-dimensional Banach space there are rows with two-element sums. The conjecture expressed in Problem 106 of the Scottish Book has thus proven dramatically wrong.

More negative statements

In finite-dimensional spaces, sum sets are always closed as affine subspaces . This property is no longer valid in infinite-dimensional spaces, as MI Ostrowskii was able to show in 1986.

The equivalence between absolute convergence and unconditional convergence is also lost in infinite-dimensional spaces, because the following Dvoretzky-Rogers theorem applies :

• Let be an infinitely dimensional Banach space. Continue with . Then there is an unconditionally convergent series with for all n. ${\displaystyle X}$ ${\displaystyle \alpha _{n}>0}$ ${\displaystyle \textstyle \sum _{n=1}^{\infty }\alpha _{n}^{2}<\infty }$ ${\displaystyle \textstyle \sum _{n=1}^{\infty }x_{n}}$ ${\displaystyle \|x_{n}\|=\alpha _{n}}$

If one chooses special , then this theorem yields the existence of an unconditionally convergent series whose summands have the norm . This series is therefore not absolutely convergent. ${\displaystyle \alpha _{n}={\tfrac {1}{n}}}$ ${\displaystyle {\tfrac {1}{n}}}$

Positive results

Despite the list of negative results above, some positive results can also be noted. From the absolute convergence also follows the unconditional convergence in infinitely dimensional spaces and this is equivalent to both the perfect convergence and the partial series convergence. Furthermore, the sum of an unconditionally convergent series is always one element.

If one adds additional requirements or one considers special spaces, then one can prove generalizations of Steinitz's rearrangement theorem:

• Be a convergent series in L ^p [0,1] , and it was . Then is . ${\displaystyle \textstyle s=\sum _{n=1}^{\infty }x_{n}}$ ${\displaystyle 1<p<\infty }$ ${\displaystyle \textstyle \sum _{n=1}^{\infty }\|x_{n}\|^{\min(2,p)}<\infty }$ ${\displaystyle SM((x_{n})_{n})=s+\Gamma ((x_{n})_{n})^{0}}$
• DV Pecherskii (1988): Let be a convergent series in a Banach space. For every rearrangement there is such a thing as converges. Then is . ${\displaystyle \textstyle s=\sum _{n=1}^{\infty }x_{n}}$ ${\displaystyle \sigma \in S({\mathbb {N} })}$ ${\displaystyle \alpha _{n}\in \{-1,1\}}$ ${\displaystyle \textstyle \sum _{n=1}^{\infty }\alpha _{n}x_{\sigma (n)}}$ ${\displaystyle SM((x_{n})_{n})=s+\Gamma ((x_{n})_{n})^{0}}$

A result by Wojciech Banaszczyk goes in a completely different direction . One can define classes of locally convex spaces that have a lot more properties in common with finite-dimensional spaces than Banach spaces; this is especially true for compactness properties. One can therefore hope to obtain generalizations of Steinitz's rearrangement theorem in such space classes, and in fact the following theorem holds:

• Let be a convergent series in a metrizable , nuclear space. Then is . ${\displaystyle \textstyle s=\sum _{n=1}^{\infty }x_{n}}$ ${\displaystyle SM((x_{n})_{n})=s+\Gamma ((x_{n})_{n})^{0}}$

swell

• W. Banaszczyk: The Steinitz theorem on rearrangement of series for nuclear spaces . Journal for pure and applied mathematics 403 (1990), 187-200.
• MI Kadets, VM Kadets: Series in Banach Spaces . Operator Theory: Advances and Applications, Vol. 94, Birkhäuser (1997), ISBN 978-3764354015 .
• MI Kadets, K. Wozniakowski: On Series Whose Permutations Have Only Two Sums . Bull. Polish Acad. Sciences Mathematics 37: 15-21 (1989).
• PA Kornilow: On the Set of Sums of a Conditionally Convergent Series of Functions . Math USSR Sbornik 65, No 1 (1990), 119-131.
• MI Ostrowskii: Domains of Sums of Conditionally Convergent Series in Banach Spaces . Teor. Functsii functional. Anal. i Prilozhen 46: 77-85 (1986).