Displaying similar documents to “On bases, compactness and weak convergence in the Banach space A p

Corrigendum to the paper “The universal Banach space with a K -suppression unconditional basis”

Taras O. Banakh, Joanna Garbulińska-Wegrzyn (2020)

Commentationes Mathematicae Universitatis Carolinae

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We observe that the notion of an almost 𝔉ℑ K -universal based Banach space, introduced in our earlier paper [1]: Banakh T., Garbulińska-Wegrzyn J., The universal Banach space with a K -suppression unconditional basis, Comment. Math. Univ. Carolin. 59 (2018), no. 2, 195–206, is vacuous for K = 1 . Taking into account this discovery, we reformulate Theorem 5.2 from [1] in order to guarantee that the main results of [1] remain valid.

The universal Banach space with a K -suppression unconditional basis

Taras O. Banakh, Joanna Garbulińska-Wegrzyn (2018)

Commentationes Mathematicae Universitatis Carolinae

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Using the technique of Fraïssé theory, for every constant K 1 , we construct a universal object 𝕌 K in the class of Banach spaces possessing a normalized K -suppression unconditional Schauder basis.

Uniqueness of unconditional basis of p ( c ) and p ( ) , 0 < p < 1

F. Albiac, C. Leránoz (2002)

Studia Mathematica

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We prove that the quasi-Banach spaces p ( c ) and p ( ) (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 ( c ) and ℓ₁(ℓ₂). They used duality techniques which are not available in the non-locally convex case.

Generalization of the weak amenability on various Banach algebras

Madjid Eshaghi Gordji, Ali Jabbari, Abasalt Bodaghi (2019)

Mathematica Bohemica

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The generalized notion of weak amenability, namely ( ϕ , ψ ) -weak amenability, where ϕ , ψ are continuous homomorphisms on a Banach algebra 𝒜 , was introduced by Bodaghi, Eshaghi Gordji and Medghalchi (2009). In this paper, the ( ϕ , ψ ) -weak amenability on the measure algebra M ( G ) , the group algebra L 1 ( G ) and the Segal algebra S 1 ( G ) , where G is a locally compact group, are studied. As a typical example, the ( ϕ , ψ ) -weak amenability of a special semigroup algebra is shown as well.

Convergence of greedy approximation I. General systems

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 T ε ( x ) : = j D ε ( x ) e * j ( x ) e j , where D ε ( x ) : = j : | e * j ( x ) | ε . We study a generalized version of T ε that we call the weak thresholding approximation. We modify the T ε ( x ) in the following way. For ε > 0, t ∈ (0,1) we set D t , ε ( x ) : = j : t ε | e * j ( x ) | < ε and consider...

Three-space problems and bounded approximation properties

Wolfgang Lusky (2003)

Studia Mathematica

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Let R n = 1 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 p -space, then both X and A have bases. We apply these results to show that the spaces C Λ = s p a n ¯ z k : k Λ C ( ) and L Λ = s p a n ¯ z k : k Λ L ( ) have bases whenever Λ ⊂ ℤ and ℤ∖Λ is a Sidon set.

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

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 C k ( X ) is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every k -space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space C k ( X ) is Ascoli iff C k ( X ) is a k -space iff X is locally compact. Moreover, C k ( X ) endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability...

Δ -weak character amenability of certain Banach algebras

Hamid Sadeghi (2019)

Archivum Mathematicum

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In this paper we investigate Δ -weak character amenability of certain Banach algebras such as projective tensor product A ^ B and Lau product A × θ B , where A and B are two arbitrary Banach algebras and θ Δ ( B ) , the character space of B . We also investigate Δ -weak character amenability of abstract Segal algebras and module extension Banach algebras.

Addendum to "Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces" (Colloq. Math. 127 (2012), 105-109)

Aydin Sh. Shukurov (2014)

Colloquium Mathematicae

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It is well known that if φ(t) ≡ t, then the system φ ( t ) n = 0 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 φ ( t ) n = 0 is a basis in some Lebesgue space L p . The aim of this short note is to show that the answer to this question is negative.

On Some Properties of Separately Increasing Functions from [0,1]ⁿ into a Banach Space

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 f : [ 0 , 1 ] m X 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 f : [ 0 , 1 ] m X with respect to any norming subset there exists a separately increasing function g : [ 0 , 1 ] m such that the sets of...

Reflexivity and approximate fixed points

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 x n k for which s u p f S X * l i m s u p k f ( x n k ) is attained at some f in the dual unit sphere S X * . A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every f S X * , there exists g S X * such that l i m s u p n f ( x ) < l i m i n f n g ( x ) . Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded...

Sequentially Right Banach spaces of order p

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 p , and those defined by the dual property, the sequentially Right * Banach spaces of order p for 1 p . These classes of Banach spaces are characterized by the notions of L p -limited sets in the corresponding dual space and R p * subsets of the involved Banach space, respectively. In particular, we investigate whether the injective tensor product of a Banach space X and a reflexive Banach...

The Lindelöf property in Banach spaces

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 M D the following four conditions are equivalent: (i) K is fragmented by d D , where, for each S ⊂ D, d S ( x , y ) = s u p ϱ ( x ( t ) , y ( t ) ) : t S . (ii) For each countable subset...

The subspace of weak P -points of *

Salvador García-Ferreira, Y. F. Ortiz-Castillo (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let W be the subspace of * consisting of all weak P -points. It is not hard to see that W is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that W is a p -pseudocompact space for all p * .

On the Aronszajn property for integral equations in Banach space

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 J = [ 0 , a ] of a solution x with values in a Banach space E , belonging to the class L p ( J , E ) , p > 1 . Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

Structure of Cesàro function spaces: a survey

Sergey V. Astashkin, Lech Maligranda (2014)

Banach Center Publications

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Geometric structure of Cesàro function spaces C e s p ( I ) , 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 C e s p [ 0 , 1 ] contains isomorphic and complemented copies of l q -spaces, show that Cesàro function spaces fail the fixed point property, give a description of subspaces generated by Rademacher functions in spaces C e s p [ 0 , 1 ] .

On the class of positive disjoint weak p -convergent operators

Abderrahman Retbi (2024)

Mathematica Bohemica

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We introduce and study the disjoint weak p -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 p -convergent operators. Next, we examine the relationship between disjoint weak p -convergent operators and disjoint p -convergent operators. Finally, we characterize order bounded disjoint weak p -convergent operators...

Every separable Banach space has a basis with uniformly controlled permutations

Paolo Terenzi

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There exists a universal control sequence p ̅ ( m ) m = 1 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, x = n = 1 x π ( n ) * ( x ) x π ( n ) where π(n) is a permutation of n which depends on x but is uniformly controlled by p ̅ ( m ) m = 1 , that is, n n = 1 m π ( n ) n = 1 p ̅ ( m ) n n = 1 p ̅ ( m + 1 ) for each m.