Lévy–Steinitz theorem

The Lévy–Steinitz theorem characterizes all sums obtainable by rearranging a convergent series in a finite-dimensional real vector space. Those sums form an affine subspace determined by the series’ conditionally convergent directions.

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Rearrangement theorem: A result describing how rearranging the terms of a series affects its convergence and sum. The theorem extends rearrangement results from real series to finite-dimensional vector series.
Riemann rearrangement theorem: A theorem stating that a conditionally convergent real series can be rearranged to converge to any prescribed real number. It is the one-dimensional model for the theorem’s characterization of vector-series sums.
Ernst Steinitz: A German mathematician whose work included the structure of sums of rearranged vector series. His name is attached to the theorem’s vector-series characterization.
Riemann series theorem: A theorem characterizing the sums obtainable by rearranging a conditionally convergent real series. It is the scalar consequence recovered when the vector space has dimension one.
Unconditional convergence: Convergence of a series whose sum remains unchanged under every permutation of its terms. Unconditional convergence is the special case where rearrangement cannot produce other sums.
Finite-dimensional vector space: A vector space with a finite basis. Finite dimensionality makes the set of rearranged sums an affine subspace.
Paul Lévy: A French mathematician known for foundational work in probability and analysis. His work on rearranged series contributed the theorem’s other eponym.
Steinitz lemma: A result in finite-dimensional vector spaces that bounds rearrangements of vectors with a prescribed total sum. It provides a finite-dimensional tool related to controlling rearranged vector sums.
Conditional convergence: Convergence of a series that fails to converge absolutely. The theorem’s nontrivial geometry comes from directions in which convergence is conditional.
Convergent series: A sequence of partial sums that approaches a finite limit. The theorem starts with an already convergent vector series.