کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4614386 1339288 2016 26 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Direct sums and products in topological groups and vector spaces
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات آنالیز ریاضی
پیش نمایش صفحه اول مقاله
Direct sums and products in topological groups and vector spaces
چکیده انگلیسی

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⊆AF⊆A such that the subgroup generated by A∖FA∖F is contained in U; (ii) absolutely summable   if, for every family {za:a∈A}{za:a∈A} of integer numbers, there exists g∈Gg∈G such that the net {∑a∈Fzaa:F⊆A is finite}{∑a∈Fzaa:F⊆A is finite} converges to g; (iii) topologically independent   provided that 0∉A0∉A and for every neighbourhood W of 0 there exists a neighbourhood V   of 0 such that, for every finite set F⊆AF⊆A and each set {za:a∈F}{za:a∈F} of integers, ∑a∈Fzaa∈V∑a∈Fzaa∈V implies that zaa∈Wzaa∈W for all a∈Fa∈F. We prove that: (1) an abelian topological group contains a direct product (direct sum) of κ-many non-trivial topological groups if and only if it contains a topologically independent, absolutely (Cauchy) summable subset of cardinality κ  ; (2) a topological vector space contains R(N)R(N) as its subspace if and only if it has an infinite absolutely Cauchy summable set; (3) a topological vector space contains RNRN as its subspace if and only if it has an RNRN multiplier convergent series of non-zero elements. We answer a question of Hušek and generalize results by Bessaga–Pelczynski–Rolewicz, Dominguez–Tarieladze and Lipecki.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Mathematical Analysis and Applications - Volume 437, Issue 2, 15 May 2016, Pages 1257–1282
نویسندگان
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