## Abstract

Let Σ_{n₤w}^{h(n)} be a conditionally convergent series in a real Banach space B. Let S(h) denote the set of sums of the convergent rearrangements of this series. A well-known theorem of Riemann states that S(h) - B if B — R, the reals. A generalization of Riemann's Theorem, due independently to Levy [L] and Steinitz [S], states that if £ is finite dimensional, then S(h) is a linear manifold in B of dimension > 0. Another generalization of Riemann's Theorem [M] can be stated as an instance of the Levy-Steinitz Theorem in the Banach space of regulated real functions on the unit interval I. This instance generalizes to the Banach space of regulated B-valued functions on /, where B is finite dimensional, implying a generalization of the Levy-Steinitz Theorem.

Original language | English |
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Pages (from-to) | 483-491 |

Number of pages | 9 |

Journal | Transactions of the American Mathematical Society |

Volume | 246 |

DOIs | |

State | Published - Dec 1978 |

## Keywords

- Conditionally convergent series
- Finite dimensional Banach space
- Rearrangement of terms

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics