Let Σn₤wh(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.
- Conditionally convergent series
- Finite dimensional Banach space
- Rearrangement of terms
ASJC Scopus subject areas
- Mathematics (all)
- Applied Mathematics