Norm continuity of weakly quasi-continuous mappings

Alireza Kamel Mirmostafaee

Displaying similar documents to “Norm continuity of weakly quasi-continuous mappings”

Conditionality constants of quasi-greedy bases in super-reflexive Banach spaces

F. Albiac, J. L. Ansorena, G. Garrigós, E. Hernández, M. Raja (2015)

Studia Mathematica

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We show that in a super-reflexive Banach space, the conditionality constants $k_{N} (ℬ)$ of a quasi-greedy basis ℬ grow at most like $O ({(l o g N)}^{1 - ε})$ for some 0 < ε < 1. This extends results by the third-named author and Wojtaszczyk (2014), where this property was shown for quasi-greedy bases in $L_{p}$ for 1 < p < ∞. We also give an example of a quasi-greedy basis ℬ in a reflexive Banach space with $k_{N} (ℬ) \approx l o g N$ .

Boundedness of sublinear operators in Triebel-Lizorkin spaces via atoms

Liguang Liu, Dachun Yang (2009)

Studia Mathematica

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Let s ∈ ℝ, p ∈ (0,1] and q ∈ [p,∞). It is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from the Triebel-Lizorkin space $Ḟ_{p, q}^{s} (ℝ ⁿ)$ to a quasi-Banach space ℬ if and only if sup ${| | T (a) | |}_{ℬ}$ : a is an infinitely differentiable (p,q,s)-atom of $Ḟ_{p, q}^{s} (ℝ ⁿ)$ < ∞, where the (p,q,s)-atom of $Ḟ_{p, q}^{s} (ℝ ⁿ)$ is as defined by Han, Paluszyński and Weiss.

Isomorphisms of some reflexive algebras

Jiankui Li, Zhidong Pan (2008)

Studia Mathematica

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Suppose ℒ₁ and ℒ₂ are subspace lattices on complex separable Banach spaces X and Y, respectively. We prove that under certain lattice-theoretic conditions every isomorphism from algℒ₁ to algℒ₂ is quasi-spatial; in particular, if a subspace lattice ℒ of a complex separable Banach space X contains a sequence $E_{i}$ such that $(E_{i}) ₋ \neq X$ , $E_{i} \subseteq E_{i + 1}$ , and $⋁_{i = 1}^{\infty} E_{i} = X$ then every automorphism of algℒ is quasi-spatial.

Uniqueness of unconditional basis of $ℓ_{p} (c ₀)$ and $ℓ_{p} (ℓ ₂)$ , 0 < p < 1

F. Albiac, C. Leránoz (2002)

Studia Mathematica

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We prove that the quasi-Banach spaces $ℓ_{p} (c ₀)$ and $ℓ_{p} (ℓ ₂)$ (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 $ℓ ₁ (c ₀)$ and ℓ₁(ℓ₂). They used duality techniques which are not available in the non-locally convex case.

On the inclusions of $X^{Φ}$ spaces

Seyyed Mohammad Tabatabaie, Alireza Bagheri Salec (2023)

Mathematica Bohemica

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We give some equivalent conditions (independent from the Young functions) for inclusions between some classes of $X^{Φ}$ spaces, where $Φ$ is a Young function and $X$ is a quasi-Banach function space on a $σ$ -finite measure space $(Ω, 𝒜, μ)$ .

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 $1 \leq p \leq \infty$ . 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...

Zero-sums of length kq in $ℤ_{q}^{d}$

Silke Kubertin (2005)

Acta Arithmetica

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From restricted type to strong type estimates on quasi-Banach rearrangement invariant spaces

María Carro, Leonardo Colzani, Gord Sinnamon (2007)

Studia Mathematica

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Let X be a quasi-Banach rearrangement invariant space and let T be an (ε,δ)-atomic operator for which a restricted type estimate of the form $∥ T χ_{E} ∥_{X} \leq D (| E |)$ for some positive function D and every measurable set E is known. We show that this estimate can be extended to the set of all positive functions f ∈ L¹ such that ${| | f | |}_{\infty} \leq 1$ , in the sense that $∥ T f ∥_{X} \leq D (| | f | | ₁)$ . This inequality allows us to obtain strong type estimates for T on several classes of spaces as soon as some information about the galb of the space X is known....

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

Some results on quasi-t-dual Baer modules

Rachid Tribak, Yahya Talebi, Mehrab Hosseinpour (2023)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a ring and let $M$ be an $R$ -module with $S = {End}_{R} (M)$ . Consider the preradical $\bar{Z}$ for the category of right $R$ -modules Mod- $R$ introduced by Y. Talebi and N. Vanaja in 2002 and defined by $\bar{Z} (M) = ⋂ {U \leq M : M / U$ is small in its injective hull $}$ . The module $M$ is called quasi-t-dual Baer if $\sum_{ϕ \in ℑ} ϕ ({\bar{Z}}^{2} (M))$ is a direct summand of $M$ for every two-sided ideal $ℑ$ of $S$ , where ${\bar{Z}}^{2} (M) = \bar{Z} (\bar{Z} (M))$ . In this paper, we show that $M$ is quasi-t-dual Baer if and only if ${\bar{Z}}^{2} (M)$ is a direct summand of $M$ and ${\bar{Z}}^{2} (M)$ is a quasi-dual Baer module. It is also shown that any direct...

On asymptotically symmetric Banach spaces

M. Junge, D. Kutzarova, E. Odell (2006)

Studia Mathematica

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A Banach space X is asymptotically symmetric (a.s.) if for some C < ∞, for all m ∈ ℕ, for all bounded sequences ${(x_{j}^{i})}_{j = 1}^{\infty} \subseteq X$ , 1 ≤ i ≤ m, for all permutations σ of 1,...,m and all ultrafilters ₁,...,ₘ on ℕ, $l i m_{n ₁, ₁} . . . l i m_{n ₘ, ₘ} | | \sum_{i = 1}^{m} x_{n_{i}}^{i} | | \leq C l i m_{n_{σ (1)},_{σ (1)}} . . . l i m_{n_{σ (m)},_{σ (m)}} | | \sum_{i = 1}^{m} x_{n_{i}}^{i} | |$ . We investigate a.s. Banach spaces and several natural variations. X is weakly a.s. (w.a.s.) if the defining condition holds when restricted to weakly convergent sequences ${(x_{j}^{i})}_{j = 1}^{\infty}$ . Moreover, X is w.n.a.s. if we restrict the condition further to normalized weakly null sequences. If X is a.s. then...

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 $s u p_{f \in S_{X *}} l i m s u p_{k \to \infty} f (x_{n_{k}})$ 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 $f \in S_{X *}$ , there exists $g \in S_{X *}$ such that $l i m s u p_{n \to \infty} f (x ₙ) < l i m i n f_{n \to \infty} g (x ₙ)$ . Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded...

Geometry of Banach spaces and biorthogonal systems

S. Dilworth, Maria Girardi, W. Johnson (2000)

Studia Mathematica

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A separable Banach space X contains $ℓ_{1}$ isomorphically if and only if X has a bounded fundamental total $w c_{0} *$ -stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded fundamental total $w c_{0} *$ -biorthogonal system.

Order convexity and concavity of Lorentz spaces $Λ_{p, w}$ , 0 < p < ∞

Anna Kamińska, Lech Maligranda (2004)

Studia Mathematica

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We study order convexity and concavity of quasi-Banach Lorentz spaces $Λ_{p, w}$ , where 0 < p < ∞ and w is a locally integrable positive weight function. We show first that $Λ_{p, w}$ contains an order isomorphic copy of $l^{p}$ . We then present complete criteria for lattice convexity and concavity as well as for upper and lower estimates for $Λ_{p, w}$ . We conclude with a characterization of the type and cotype of $Λ_{p, w}$ in the case when $Λ_{p, w}$ is a normable space.

On the $k$ -polygonal numbers and the mean value of Dedekind sums

Jing Guo, Xiaoxue Li (2016)

Czechoslovak Mathematical Journal

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For any positive integer $k \geq 3$ , it is easy to prove that the $k$ -polygonal numbers are $a_{n} (k) = (2 n + n (n - 1) (k - 2)) / 2$ . The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet $L$ -functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums $S (a_{n} (k) {\bar{a}}_{m} (k), p)$ for $k$ -polygonal numbers with $1 \leq m, n \leq p - 1$ , and give an interesting computational formula for it.

Norm continuity of pointwise quasi-continuous mappings

Alireza Kamel Mirmostafaee (2018)

Mathematica Bohemica

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Let $X$ be a Baire space, $Y$ be a compact Hausdorff space and $ϕ : X \to C_{p} (Y)$ be a quasi-continuous mapping. For a proximal subset $H$ of $Y \times Y$ we will use topological games $𝒢_{1} (H)$ and $𝒢_{2} (H)$ on $Y \times Y$ between two players to prove that if the first player has a winning strategy in these games, then $ϕ$ is norm continuous on a dense $G_{δ}$ subset of $X$ . It follows that if $Y$ is Valdivia compact, each quasi-continuous mapping from a Baire space $X$ to $C_{p} (Y)$ is norm continuous on a dense $G_{δ}$ subset of $X$ .

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, $d_{S} (x, y) = s u p ϱ (x (t), y (t)) : t \in S$ . (ii) For each countable subset...

A note on a class of homeomorphisms between Banach spaces

Piotr Fijałkowski (2005)

Colloquium Mathematicae

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This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form $F (x) : = F ̃ x^{(2 n + 1)}$ where $F ̃ : X^{2 n + 1} \to Y$ is a continuous (2n+1)-linear operator.

On the H-property and rotundity of Cesàro direct sums of Banach spaces

Saard Youyen, Suthep Suantai (2008)

Banach Center Publications

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In this paper, we define the direct sum ${(⨁_{i = 1}^{n} X_{i})}_{c e s_{p}}$ of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that ${(⨁_{i = 1}^{n} X_{i})}_{c e s_{p}}$ has the H-property if and only if each $X_{i}$ has the H-property, and ${(⨁_{i = 1}^{n} X_{i})}_{c e s_{p}}$ has the Schur property if and only if each $X_{i}$ has the Schur property. Moreover, we also show that ${(⨁_{i = 1}^{n} X_{i})}_{c e s_{p}}$ is rotund if and only if each $X_{i}$ is rotund.

Application of $(L)$ sets to some classes of operators

Kamal El Fahri, Nabil Machrafi, Jawad H&#039;michane, Aziz Elbour (2016)

Mathematica Bohemica

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The paper contains some applications of the notion of $Ł$ sets to several classes of operators on Banach lattices. In particular, we introduce and study the class of order $(L)$ -Dunford-Pettis operators, that is, operators from a Banach space into a Banach lattice whose adjoint maps order bounded subsets to an $(L)$ sets. As a sequence characterization of such operators, we see that an operator $T : X \to E$ from a Banach space into a Banach lattice is order $Ł$ -Dunford-Pettis, if and only if $| T (x_{n}) | \to 0$ for $σ (E, E^{'})$ for every...

On Banach spaces C(K) isomorphic to c₀(Γ)

Witold Marciszewski (2003)

Studia Mathematica

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We give a characterization of compact spaces K such that the Banach space C(K) is isomorphic to the space c₀(Γ) for some set Γ. As an application we show that there exists an Eberlein compact space K of weight $ω_{ω}$ and with the third derived set $K^{(3)}$ empty such that the space C(K) is not isomorphic to any c₀(Γ). For this compactum K, the spaces C(K) and $c ₀ (ω_{ω})$ are examples of weakly compactly generated (WCG) Banach spaces which are Lipschitz isomorphic but not isomorphic.

On the compact approximation property

Vegard Lima, Åsvald Lima, Olav Nygaard (2004)

Studia Mathematica

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We show that a Banach space X has the compact approximation property if and only if for every Banach space Y and every weakly compact operator T: Y → X, the space = S ∘ T: S compact operator on X is an ideal in = span(,T) if and only if for every Banach space Y and every weakly compact operator T: Y → X, there is a net $(S_{γ})$ of compact operators on X such that $s u p_{γ} | | S_{γ} T | | \leq | | T | |$ and $S_{γ} \to I_{X}$ in the strong operator topology. Similar results for dual spaces are also proved.

Linearization of isometric embedding on Banach spaces

Yu Zhou, Zihou Zhang, Chunyan Liu (2015)

Studia Mathematica

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Let X,Y be Banach spaces, f: X → Y be an isometry with f(0) = 0, and $T : \bar{s p a n} (f (X)) \to X$ be the Figiel operator with $T \circ f = I d_{X}$ and ||T|| = 1. We present a sufficient and necessary condition for the Figiel operator T to admit a linear isometric right inverse. We also prove that such a right inverse exists when $\bar{s p a n} (f (X))$ is weakly nearly strictly convex.

Limited $p$ -converging operators and relation with some geometric properties of Banach spaces

Mohammad B. Dehghani, Seyed M. Moshtaghioun (2021)

Commentationes Mathematicae Universitatis Carolinae

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By using the concepts of limited $p$ -converging operators between two Banach spaces $X$ and $Y$ , $L_{p}$ -sets and $L_{p}$ -limited sets in Banach spaces, we obtain some characterizations of these concepts relative to some well-known geometric properties of Banach spaces, such as $*$ -Dunford–Pettis property of order $p$ and Pelczyński’s property of order $p$ , $1 \leq p < \infty$ .