Productivity of sequences in non-abelian topological groups

Spěvák, Jan

doi:10.1016/j.topol.2015.05.084

Cited by 2 publications

(3 citation statements)

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“…i=0 a i by the definition of a i . Combining this with the linearity of π A a k , (20) and n ≥ k, we get 21), the map π A a k is discontinuous. Remark 9.5.…”

Section: Applications To Metric Nss Groupsmentioning

confidence: 79%

“…By Lemma 9.3, it suffices to show that the projection π A a k is discontinuous for every k ∈ N. Fix a k ∈ N. It follows from the definition of π A a k that (20) π A a k (a k ) = a k and π A a k (a i ) = 0 for all i ∈ N with i = k. Let n ∈ N and n ≥ k. Note that e n = n i=0 a i by the definition of a i . Combining this with the linearity of π A a k , (20) and n ≥ k, we get…”

Section: Continuity Of Finite-dimensional Projections In Topological mentioning

confidence: 99%

“…Further, from Proposition 6.4 it follows that it is f ω -Cauchy summable if and only if it is absolutely Cauchy summable. For other versions of f ω -summability see [5,20]. Remark 6.8.…”

Section: Proof Items (Ii) and (Iii) Are Obviously Equivalent And (Ii)...mentioning

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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“…i=0 a i by the definition of a i . Combining this with the linearity of π A a k , (20) and n ≥ k, we get 21), the map π A a k is discontinuous. Remark 9.5.…”

Section: Applications To Metric Nss Groupsmentioning

confidence: 79%

Section: Continuity Of Finite-dimensional Projections In Topological mentioning

confidence: 99%