On ultra-weak convergence in
Donald E. Myers (1976)
Annales Polonici Mathematici
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Donald E. Myers (1976)
Annales Polonici Mathematici
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Taras O. Banakh, Joanna Garbulińska-Wegrzyn (2020)
Commentationes Mathematicae Universitatis Carolinae
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We observe that the notion of an almost -universal based Banach space, introduced in our earlier paper [1]: Banakh T., Garbulińska-Wegrzyn J., The universal Banach space with a -suppression unconditional basis, Comment. Math. Univ. Carolin. 59 (2018), no. 2, 195–206, is vacuous for . Taking into account this discovery, we reformulate Theorem 5.2 from [1] in order to guarantee that the main results of [1] remain valid.
Taras O. Banakh, Joanna Garbulińska-Wegrzyn (2018)
Commentationes Mathematicae Universitatis Carolinae
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Using the technique of Fraïssé theory, for every constant , we construct a universal object in the class of Banach spaces possessing a normalized -suppression unconditional Schauder basis.
F. Albiac, C. Leránoz (2002)
Studia Mathematica
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We prove that the quasi-Banach spaces and (0 < p < 1) have a unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss and Tzafriri have previously proved that the same is true for the respective Banach envelopes and ℓ₁(ℓ₂). They used duality techniques which are not available in the non-locally convex case.
Madjid Eshaghi Gordji, Ali Jabbari, Abasalt Bodaghi (2019)
Mathematica Bohemica
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The generalized notion of weak amenability, namely -weak amenability, where are continuous homomorphisms on a Banach algebra , was introduced by Bodaghi, Eshaghi Gordji and Medghalchi (2009). In this paper, the -weak amenability on the measure algebra , the group algebra and the Segal algebra , where is a locally compact group, are studied. As a typical example, the -weak amenability of a special semigroup algebra is shown as well.
S. V. Konyagin, V. N. Temlyakov (2003)
Studia Mathematica
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We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as , where . We study a generalized version of that we call the weak thresholding approximation. We modify the in the following way. For ε > 0, t ∈ (0,1) we set and consider...
Wolfgang Lusky (2003)
Studia Mathematica
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Let be a commuting approximating sequence of the Banach space X leaving the closed subspace A ⊂ X invariant. Then we prove three-space results of the following kind: If the operators Rₙ induce basis projections on X/A, and X or A is an -space, then both X and A have bases. We apply these results to show that the spaces and have bases whenever Λ ⊂ ℤ and ℤ∖Λ is a Sidon set.
Larry Kitchens, Charles Swartz (1974)
Colloquium Mathematicae
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S. Gabriyelyan, J. Kąkol, G. Plebanek (2016)
Studia Mathematica
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Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every -space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space is Ascoli iff is a -space iff X is locally compact. Moreover, endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability...
Hamid Sadeghi (2019)
Archivum Mathematicum
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In this paper we investigate -weak character amenability of certain Banach algebras such as projective tensor product and Lau product , where and are two arbitrary Banach algebras and , the character space of . We also investigate -weak character amenability of abstract Segal algebras and module extension Banach algebras.
Sreela Gangopadhyay (1990)
Colloquium Mathematicae
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Aydin Sh. Shukurov (2014)
Colloquium Mathematicae
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It is well known that if φ(t) ≡ t, then the system is not a Schauder basis in L₂[0,1]. It is natural to ask whether there is a function φ for which the power system is a basis in some Lebesgue space . The aim of this short note is to show that the answer to this question is negative.
Aydin Sh. Shukurov (2012)
Colloquium Mathematicae
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A necessary condition for Kostyuchenko type systems and system of powers to be a basis in (1 ≤ p < +∞) spaces is obtained. In particular, we find a necessary condition for a Kostyuchenko system to be a basis in (1 ≤ p < +∞).
Artur Michalak (2014)
Bulletin of the Polish Academy of Sciences. Mathematics
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We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function with respect to any norming subset there exists a separately increasing function such that the sets of...
Eva Matoušková, Simeon Reich (2003)
Studia Mathematica
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A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence for which is attained at some f in the dual unit sphere . A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every , there exists such that . Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded...
Mahdi Dehghani, Mohammad B. Dehghani, Mohammad S. Moshtaghioun (2020)
Commentationes Mathematicae Universitatis Carolinae
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We introduce and study two new classes of Banach spaces, the so-called sequentially Right Banach spaces of order , and those defined by the dual property, the sequentially Right Banach spaces of order for . These classes of Banach spaces are characterized by the notions of -limited sets in the corresponding dual space and subsets of the involved Banach space, respectively. In particular, we investigate whether the injective tensor product of a Banach space and a reflexive Banach...
B. Cascales, I. Namioka, J. Orihuela (2003)
Studia Mathematica
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A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space the following four conditions are equivalent: (i) K is fragmented by , where, for each S ⊂ D, . (ii) For each countable subset...
Salvador García-Ferreira, Y. F. Ortiz-Castillo (2015)
Commentationes Mathematicae Universitatis Carolinae
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Let be the subspace of consisting of all weak -points. It is not hard to see that is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that is a -pseudocompact space for all .
Stanisław Szufla (1989)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
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For the integral equation (1) below we prove the existence on an interval of a solution with values in a Banach space , belonging to the class , . Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.
Sergey V. Astashkin, Lech Maligranda (2014)
Banach Center Publications
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Geometric structure of Cesàro function spaces , where I = [0,1] and [0,∞), is investigated. Among other matters we present a description of their dual spaces, characterize the sets of all q ∈ [1,∞] such that contains isomorphic and complemented copies of -spaces, show that Cesàro function spaces fail the fixed point property, give a description of subspaces generated by Rademacher functions in spaces .
Abderrahman Retbi (2024)
Mathematica Bohemica
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We introduce and study the disjoint weak -convergent operators in Banach lattices, and we give a characterization of it in terms of sequences in the positive cones. As an application, we derive the domination and the duality properties of the class of positive disjoint weak -convergent operators. Next, we examine the relationship between disjoint weak -convergent operators and disjoint -convergent operators. Finally, we characterize order bounded disjoint weak -convergent operators...
Paolo Terenzi
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There exists a universal control sequence of increasing positive integers such that: Every infinite-dimensional separable Banach space X has a biorthogonal system xₙ,xₙ* with ||xₙ|| = 1 and ||xₙ*|| < K for each n such that, for each x ∈ X, where π(n) is a permutation of n which depends on x but is uniformly controlled by , that is, for each m.
W. Szczotka (2006)
Applicationes Mathematicae
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For each n ≥ 1, let and be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let , , t ≥ 0, and . The main result gives conditions under which the weak convergence , where X is a Lévy process, implies and , where and are mutually independent Lévy processes and .
Tuomas Hytönen, Assaf Naor (2013)
Studia Mathematica
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Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies , where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and and are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by , we show that for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...
Evgeniy Pustylnik (2001)
Studia Mathematica
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We give a full solution of the following problems concerning the spaces : (i) to what extent two functions φ and ψ should be different in order to ensure that for any nontrivial Banach couple X⃗; (ii) when an embedding can (or cannot) be dense; (iii) which Banach space can be regarded as an -space for some (unknown beforehand) Banach couple X⃗.
Christina Brech, Piotr Koszmider (2014)
Fundamenta Mathematicae
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This paper is concerned with the isomorphic structure of the Banach space and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that does not have an orthogonal -decomposition, that is, it is not of the form for any Banach space X. The main local result is that it is consistent that does not embed isomorphically into , where is the cardinality of the continuum,...