Steinitz rearrangement theorem

The Steinitz rearrangement theorem (after Ernst Steinitz ) is a theorem from mathematical analysis that deals with the rearrangement of series . While arbitrary rearrangements within finite sums due to the commutative law and the associative law have no influence on the result of the summation, this is no longer guaranteed with infinite sums. The Steinitz rearrangement theorem discussed here makes a statement about the structure of the set of sums that can be formed by rearrangement. He generalizes the Riemann rearrangement theorem , which applies to real series, to series im . ${\displaystyle {\mathbb {R} ^{m}}}$

Convergence terms for series

→ Main article : Rearrangement of rows

In , as in real numbers, one can speak of convergence, because the usual Euclidean norm provides a concept of distance. ${\displaystyle {\mathbb {R} ^{m}}}$

Let it now be a sequence of vectors im . If the limit of the partial sums im exists, one writes for this limit and says that the series is convergent . Note that the same name is used for the range and its limit. ${\displaystyle (a_{n})_{n}}$ ${\displaystyle {\mathbb {R} ^{m}}}$ ${\displaystyle \textstyle \lim _{N\rightarrow \infty }\sum _{n=1}^{N}a_{n}}$ ${\displaystyle {\mathbb {R} ^{m}}}$ ${\displaystyle \textstyle \sum _{n=1}^{\infty }a_{n}}$ ${\displaystyle \textstyle \sum _{n=1}^{\infty }a_{n}}$

Each permutation defines a rearrangement by going from sequence to sequence . One calls a convergent rearrangement of the series when the rearranged series converges. The series is said to be unconditionally convergent if every rearrangement of the series is convergent. ${\displaystyle \sigma \colon {\mathbb {N} }\rightarrow {\mathbb {N} }}$ ${\displaystyle (a_{n})_{n}}$ ${\displaystyle (a_{\sigma (n)})_{n}}$ ${\displaystyle \sigma }$ ${\displaystyle \textstyle \sum _{n=1}^{\infty }a_{\sigma (n)}}$ ${\displaystyle \textstyle \sum _{n=1}^{\infty }a_{n}}$

The series is called conditionally convergent if it is convergent, but not necessarily convergent. Finally, the series is called absolutely convergent if holds. ${\displaystyle \textstyle \sum _{n=1}^{\infty }a_{n}}$ ${\displaystyle \textstyle \sum _{n=1}^{\infty }\|a_{n}\|<\infty }$

Convergence functionals

A linear functional is called a convergence functional for the sequence , if is. So is z. B. the null functional is a convergence functional for each sequence. It is easy to consider that the set of all convergence functionals is a subspace in the dual space , i.e. H. in the space of linear functionals, is. This subspace of the convergence functionals is denoted by, the annihilator of by . ${\displaystyle f\colon {\mathbb {R} ^{m}}\rightarrow {\mathbb {R} }}$ ${\displaystyle (a_{n})_{n}}$ ${\displaystyle \textstyle \sum _{n=1}^{\infty }|f(a_{n})|<\infty }$ ${\displaystyle \Gamma ((a_{n})_{n})}$ ${\displaystyle \Gamma ((a_{n})_{n})}$ ${\displaystyle \Gamma ((a_{n})_{n})^{0}}$

Steinitz theorem

Let it be a convergent series. Then agrees with the affine subspace . ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ ${\displaystyle \left\{\sum _{n=1}^{\infty }a_{\sigma (n)}{\Bigg |}\sigma \ {\text{konvergente Umordnung}}\right\}}$ ${\displaystyle \sum _{n=1}^{\infty }a_{n}\,+\,\Gamma ((a_{n})_{n})^{0}}$

Addition: If this affine space consists of more than one point, there are non-convergent rearrangements.

Remarks

A theorem about convergent series

With the help of Steinitz's theorem one can easily show that the following statements about a convergent series im are equivalent: ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ ${\displaystyle {\mathbb {R} ^{m}}}$

• The series is absolutely convergent.
• The series is necessarily convergent.
• ${\displaystyle \Gamma ((a_{n})_{n})^{0}\,=\,\{0\}}$.
• Each linear functional is a convergence functional for the series. ${\displaystyle {\mathbb {R} ^{m}}\rightarrow {\mathbb {R} }}$

The Riemann Reordering Theorem

→ Main article : Riemann rearrangement theorem

Since every non-empty affine subspace of either consists of a point or coincides with it , one obtains the Riemann rearrangement theorem as a special case of Steinitz's rearrangement theorem. ${\displaystyle {\mathbb {R} }}$ ${\displaystyle {\mathbb {R} }}$

The infinite dimensional case

→ Main article : Rearrangement of rows

In infinite-dimensional spaces the convergence statements made here for series no longer apply. In infinite-dimensional Banach spaces there are series with two-element sum sets. You have to make additional assumptions about the series in order to arrive at a statement like Steinitz's rearrangement theorem.

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• E. Steinitz: Conditionally convergent series and convex systems . Journal for pure and applied mathematics 143 (1913), 128-175, 144 (1914), 1-40, 146 (1915), 1-52.
• MI Kadets , VM Kadets: Series in Banach Spaces . Operator Theory: Advances and Applications, Vol. 94, Birkhäuser (1997), ISBN 978-3764354015 .
• Israel Halperin: Sums of a series, permitting rearrangements . CRMath. Acad. Sci., Soc. R. Can., 8: pp. 87-102, 1986.