Series with Commuting Terms in Topologized Semigroups

Castejón, Alberto; Corbacho, Eusebio; Tarieladze, V.

doi:10.3390/axioms10040237

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Definition 2 the Infinite Product Corresponding To A Sequence1

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2023

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“…The following statement, which in a more general setting was obtained in [9], implies that the sum range problem has an easy solution in the case of unconditional convergence.…”

Section: Definition 2 the Infinite Product Corresponding To A Sequencementioning

confidence: 85%

Permutations, Signs, and Sum Ranges

Chobanyan

Domínguez

Tarieladze

et al. 2023

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The sum range SRx;X, for a sequence x=(xn)n∈N of elements of a topological vector space X, is defined as the set of all elements s∈X for which there exists a bijection (=permutation) π:N→N, such that the sequence of partial sums (∑k=1nxπ(k))n∈N converges to s. The sum range problem consists of describing the structure of the sum ranges for certain classes of sequences. We present a survey of the results related to the sum range problem in finite- and infinite-dimensional cases. First, we provide the basic terminology. Next, we devote attention to the one-dimensional case, i.e., to the Riemann–Dini theorem. Then, we deal with spaces where the sum ranges are closed affine for all sequences, and we include some counterexamples. Next, we present a complete exposition of all the known results for general spaces, where the sum ranges are closed affine for sequences satisfying some additional conditions. Finally, we formulate two open questions.

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“…The following statement, which in a more general setting was obtained in [9], implies that the sum range problem has an easy solution in the case of unconditional convergence.…”

Section: Definition 2 the Infinite Product Corresponding To A Sequencementioning

confidence: 85%