On the separable quotient problem for Banach spaces

Ferrando, J. C.; Ka̧kol, Jerzy; Pellicer, Manuel López; Śliwa, Wiesław

doi:10.7169/facm/1704

Cited by 8 publications

(4 citation statements)

References 63 publications

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“…In addition, the structures were reformed algebraically [31][32][33][34] and topologically 35 via fuzzy metrics [36][37][38][39] . Quotient forms such as factorizable [40][41][42] , metrizable 43,44 , and separable [45][46][47] especially in Banach spaces [48][49][50][51] were developed recently. Bounded groups 52,53 , Matrix groups 54 , homotopy characteristics [55][56][57][58], and projectivity 59 on topological groups were remarkable.…”

Section: Modeling Of Robot Actions Using Digital Imagementioning

confidence: 99%

Quotient on some Generalizations of topological group

Kumar¹,

Gnanachandra²,

Priya³

2023

Baghdad Sci.J

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In this paper, we define some generalizations of topological group namely -topological group, -topological group and -topological group with illustrative examples. Also, we define grill topological group with respect to a grill. Later, we deliberate the quotient on generalizations of topological group in particular -topological group. Moreover, we model a robotic system which relays on the quotient of -topological group.

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Section: Modeling Of Robot Actions Using Digital Imagementioning

confidence: 99%

Quotient on some Generalizations of topological group

Kumar¹,

Gnanachandra²,

Priya³

2023

Baghdad Sci.J

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“…The famous Rosenthal-Lacey theorem [40], [32], see also [24,Corollary 1], asserts that for each infinite compact space K the Banach space C(K) admits a quotient map onto c 0 or ℓ 2 ; we refer the reader to a survey paper [18,Theorem 18] for a detailed discussion on the theorem. The case of C p -spaces remains however open, namely, it is still unknown whether for every infinite compact space K the space C p (K) admits a quotient map onto an infinite-dimensional metrizable space, see [29].…”

Section: Introductionmentioning

confidence: 99%

On complemented copies of the space $c_0$ in spaces $C_p(X\times Y)$

Ka̧kol¹,

Marciszewski²,

Sobota³

et al. 2020

Preprint

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Cembranos and Freniche proved that for every two infinite compact Hausdorff spaces X and Y the Banach space C(X × Y ) of continuous real-valued functions on X × Y endowed with the supremum norm contains a complemented copy of the Banach space c 0 . We extend this theorem to the class of C p -spaces, that is, we prove that for all infinite Tychonoff spaces X and Y the space C p (X ×Y ) of continuous functions on X ×Y endowed with the pointwise topology contains either a complemented copy of R ω or a complemented copy of the space (c 0 ) p = {(x n ) n∈ω ∈ R ω : x n → 0}, both endowed with the product topology. We show that the latter case holds always when X × Y is pseudocompact. On the other hand, assuming the Continuum Hypothesis (or even a weaker set-theoretic assumption), we provide an example of a pseudocompact space X such that C p (X × X) does not contain a complemented copy of (c 0 ) p .As a corollary to the first result, we show that for all infinite Tychonoff spaces X and Y the space C p (X × Y ) is linearly homeomorphic to the space C p (X × Y ) × R, although, as proved earlier by Marciszewski, there exists an infinite compact space X such that C p (X) cannot be mapped onto C p (X) × R by a continuous linear surjection. This provides a positive answer to a problem of Arkhangel'ski for spaces of the form C p (X × Y ).Our another corollary-analogous to the classical Rosenthal-Lacey theorem for Banach spaces C(X) with X compact and Hausdorff-asserts that for every infinite Tychonoff spaces X and Y the space C k (X × Y ) of continuous functions on X × Y endowed with the compact-open topology admits a quotient map onto a space isomorphic to one of the following three spaces: R ω , (c 0 ) p or c 0 .

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“…The Separable Quotient problem for Banach spaces has its roots in the 1930s and is due to Stefan Banach and Stanisław Mazur. While a positive answer is known for various classes of Banach spaces [1], such as reflexive Banach spaces, weakly compactly generated Banach spaces, and more generally Banach-like spaces [2], the general problem remains unsolved. Problem 1.…”

Section: Introductionmentioning

confidence: 99%