Multiplier Convergent Series

Swartz, Charles

doi:10.1142/9789812833884

Cited by 12 publications

(37 citation statements)

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“…Since lim n→∞ e n = 0, yet lim n→∞ π A a k (e n ) = a k = 0 = π A a k (0) by (21), the map π A a k is discontinuous. Remark 9.5.…”

Section: Continuity Of Finite-dimensional Projections In Topological mentioning

confidence: 99%

“…Combining Theorem 11.1 with Corollary 11.2, one obtains the following classical result of Bessaga, Pelczynski and Rolewicz: Recall that in the multiplier convergence theory one typically takes a fixed set F ⊆ R N of "multipliers" and calls a series ∞ n=0 a n in a topological vector space F multiplier convergent provided that the series ∞ n=0 f (n)a n converges for every f ∈ F . By varying the set F of multipliers one can obtain a fine description of the level of convergence of a given series, leading to a rich theory; see [21]. The toughest convergence condition on a series is obviously imposed by taking F to be the whole R N .…”

Section: Infinite Direct Sums and Products In Topological Vector Spacesmentioning

confidence: 99%

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Direct sums and products in topological groups and vector spaces

Dikranjan

Shakhmatov

Spěvák

2016

Journal of Mathematical Analysis and Applications

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We call a subset $A$ of an abelian topological group $G$: (i) $absolutely$ $Cauchy$ $summable$ provided that for every open neighbourhood $U$ of $0$ one can find a finite set $F\subseteq A$ such that the subgroup generated by $A\setminus F$ is contained in $U$; (ii) $absolutely$ $summable$ if, for every family $\{z_a:a\in A\}$ of integer numbers, there exists $g\in G$ such that the net $\left\{\sum_{a\in F} z_a a: F\subseteq A\mbox{ is finite}\right\}$ converges to $g$; (iii) $topologically$ $independent$ provided that $0\not \in A$ and for every neighbourhood $W$ of $0$ there exists a neighbourhood $V$ of $0$ such that, for every finite set $F\subseteq A$ and each set $\{z_a:a\in F\}$ of integers, $\sum_{a\in F}z_aa\in V$ implies that $z_aa\in W$ for all $a\in F$. We prove that: (1) an abelian topological group contains a direct product (direct sum) of $\kappa$-many non-trivial topological groups if and only if it contains a topologically independent, absolutely (Cauchy) summable subset of cardinality $\kappa$; (2) a topological vector space contains $\mathbb{R}^{(\mathbb{N})}$ as its subspace if and only if it has an infinite absolutely Cauchy summable set; (3) a topological vector space contains $\mathbb{R}^{\mathbb{N}}$ as its subspace if and only if it has an $\mathbb{R}^{(\mathbb{N})}$ multiplier convergent series of non-zero elements. We answer a question of Hu\v{s}ek and generalize results by Bessaga-Pelczynski-Rolewicz, Dominguez-Tarieladze and Lipecki

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“…Since lim n→∞ e n = 0, yet lim n→∞ π A a k (e n ) = a k = 0 = π A a k (0) by (21), the map π A a k is discontinuous. Remark 9.5.…”

Section: Continuity Of Finite-dimensional Projections In Topological mentioning

confidence: 99%

Section: Infinite Direct Sums and Products In Topological Vector Spacesmentioning

confidence: 99%

Direct sums and products in topological groups and vector spaces

Dikranjan

Shakhmatov

Spěvák

2016

Journal of Mathematical Analysis and Applications

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“…Before starting this section, we give the following propostion will be used for establishing some results of this study: is bounded [14].…”

Section: The Zweier Summability Spacementioning

confidence: 99%

On some Zweier convergent vector valued multiplier spaces

Kama

2019

Sakarya University Journal of Science

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“…We next consider a version of the Hahn-Schur Theorem for group valued series (see [AS1]; [Sw6]8.1). The usual scalar version of the theorem can be obtained easily from Case 18 (see [AS1]8.2; [Sw5]8.1).…”

Section: Applicationsmentioning

confidence: 99%

“…The case where E, F are vector spaces, G is a locally convex space and b is a bilinear map is considered in [LS3]; general versions of the Orlicz-Pettis Theorem are established and numerous applications are given. A similiar treatment is given in Chapter 4 of [Sw6], pages 73-82. Li and Cho ([LC]) have used the general abstract setting above to obtain a generalization of an Orlicz-Pettis result of Kalton; we will consider this result later.…”

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confidence: 99%