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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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.
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...
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...
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 .
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,...
Dylan Airey, Bill Mance, Joseph Vandehey (2016)
Czechoslovak Mathematical Journal
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Let be a sequence of bases with . In the case when the are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose -Cantor series expansion is both -normal and -distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of , and from this construction we can provide computable constructions of numbers with atypical normality properties. ...
Witold Marciszewski (2008)
Studia Mathematica
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We consider the class of compact spaces which are modifications of the well known double arrow space. The space is obtained from a closed subset K of the unit interval [0,1] by “splitting” points from a subset A ⊂ K. The class of all such spaces coincides with the class of separable linearly ordered compact spaces. We prove some results on the topological classification of spaces and on the isomorphic classification of the Banach spaces .
Richard Haydon (1972)
Publications du Département de mathématiques (Lyon)
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Monika Budzyńska, Aleksandra Grzesik, Mariola Kot (2017)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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In this paper we prove that for each , the Banach space can be equivalently renormed in such a way that the Banach space is LUR and has a diametrically complete set with empty interior. This result extends the Maluta theorem about existence of such a set in with the Day norm. We also show that the Banach space has the weak fixed point property for nonexpansive mappings.
Saard Youyen, Suthep Suantai (2008)
Banach Center Publications
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In this paper, we define the direct sum of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that has the H-property if and only if each has the H-property, and has the Schur property if and only if each has the Schur property. Moreover, we also show that is rotund if and only if each is rotund.
Agnieszka Kałamajska (2001)
Colloquium Mathematicae
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We study the functional , where u=(u₁, ..., uₘ) and each is constant along some subspace of ℝⁿ. We show that if intersections of the ’s satisfy a certain condition then is weakly lower semicontinuous if and only if f is Λ-convex (see Definition 1.1 and Theorem 1.1). We also give a necessary and sufficient condition on to have the equivalence: is weakly continuous if and only if f is Λ-affine.
Petr Petráček, Jiří Spurný (2016)
Commentationes Mathematicae Universitatis Carolinae
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We provide a complex version of a theorem due to Bednar and Lacey characterizing real -preduals. Hence we prove a characterization of complex -preduals via a complex barycentric mapping.
Stanisław Szufla (1989)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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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.
Amaury Lambert, Florian Simatos (2014)
Annales de l'I.H.P. Probabilités et statistiques
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We consider regenerative processes with values in some general Polish space. We define their -big excursions as excursions such that , where is some given functional on the space of excursions which can be thought of as, e.g., the length or the height of . We establish a general condition that guarantees the convergence of a sequence of regenerative processes involving the convergence of -big excursions and of their endpoints, for all in a set whose closure contains . Finally,...