Products of n open subsets in the space of continuous functions on [0,1]

Ehrhard Behrends

Displaying similar documents to “Products of n open subsets in the space of continuous functions on [0,1]”

A characterization of complex $L_{1}$ -preduals via a complex barycentric mapping

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 $L_{1}$ -preduals. Hence we prove a characterization of complex $L_{1}$ -preduals via a complex barycentric mapping.

$ω_{0}$ -categoricity of generalized products

Jan Waszkiewicz (1973)

Colloquium Mathematicae

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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.

On Relative $\gamma_{k}$ -Sets

Maddalena Bonanzinga, Filippo Cammaroto, Bruno A. Pansera (2007)

Bollettino dell'Unione Matematica Italiana

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In this note we show a relative version of $\gamma_{k}$ -set introduced and studied in [12]. We give several characterizations of this property; in particular one of the characterizations is Ramsey theoretical. Also we give a result involving a property of the corresponding mapping between function spaces.

Bicrossed products of generalized Taft algebra and group algebras

Dingguo Wang, Xiangdong Cheng, Daowei Lu (2022)

Czechoslovak Mathematical Journal

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Let $G$ be a group generated by a set of finite order elements. We prove that any bicrossed product $H_{m, d} (q) ⋈ k [G]$ between the generalized Taft algebra $H_{m, d} (q)$ and group algebra $k [G]$ is actually the smash product $H_{m, d} (q) ♯ k [G]$ . Then we show that the classification of these smash products could be reduced to the description of the group automorphisms of $G$ . As an application, the classification of $H_{m, d} (q) ⋈ k [C_{n_{1}} \times C_{n_{2}}]$ is completely presented by generators and relations, where $C_{n}$ denotes the $n$ -cyclic group.

A new function space and applications

Jean Bourgain, Haïm Brezis, Petru Mironescu (2015)

Journal of the European Mathematical Society

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We define a new function space $B$ , which contains in particular BMO, BV, and $W^{1 / p, p}$ , $1 < p < \infty$ . We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving $L^{p}$ norms of integer-valued functions in $B$ . We introduce a significant closed subspace, $B_{0}$ , of $B$ , containing in particular VMO and $W^{1 / p, p}$ , $1 \leq p < \infty$ . The above mentioned estimates imply in particular that integer-valued functions belonging to $B_{0}$ are necessarily constant. This framework provides a “common roof”...

Tensor products of higher almost split sequences in subcategories

Xiaojian Lu, Deren Luo (2023)

Czechoslovak Mathematical Journal

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We introduce the algebras satisfying the $(ℬ, n)$ condition. If $Λ$ , $Γ$ are algebras satisfying the $(ℬ, n)$ , $(ℰ, m)$ condition, respectively, we give a construction of $(m + n)$ -almost split sequences in some subcategories ${(ℬ \otimes ℰ)}^{(i_{0}, j_{0})}$ of $mod (Λ \otimes Γ)$ by tensor products and mapping cones. Moreover, we prove that the tensor product algebra $Λ \otimes Γ$ satisfies the $({(ℬ \otimes ℰ)}^{(i_{0}, j_{0})}, n + m)$ condition for some integers $i_{0}$ , $j_{0}$ ; this construction unifies and extends the work of A. Pasquali (2017), (2019).

$ℓ_{\infty}$ -sums and the Banach space $ℓ_{\infty} / c ₀$

Christina Brech, Piotr Koszmider (2014)

Fundamenta Mathematicae

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This paper is concerned with the isomorphic structure of the Banach space $ℓ_{\infty} / c ₀$ 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 $ℓ_{\infty} / c ₀$ does not have an orthogonal $ℓ_{\infty}$ -decomposition, that is, it is not of the form $ℓ_{\infty} (X)$ for any Banach space X. The main local result is that it is consistent that $ℓ_{\infty} (c ₀ ())$ does not embed isomorphically into $ℓ_{\infty} / c ₀$ , where is the cardinality of the continuum,...

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.

Some theorems of Korovkin type

Tomoko Hachiro, Takateru Okayasu (2003)

Studia Mathematica

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We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ} (X)$ ) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ} (Y)$ ), and $ϕ_{\infty}$ a linear isometry from M into C(Y) (resp., $C_{ℝ} (Y)$ ). We show, under the assumption that $Π_{N} \subset Π_{T}$ , where $Π_{N}$ is...

Controlling products of currents by higher powers of plurisubharmonic functions

Ahmad K. Al Abdulaali, Hassine El Mir (2020)

Czechoslovak Mathematical Journal

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We discuss the existence of the current $g^{k} T$ , $k \in ℕ$ for positive and closed currents $T$ and unbounded plurisubharmonic functions $g$ . Furthermore, a new type of weighted Lelong number is introduced under the name of weight $k$ Lelong number.

$Σ_{s}$ -products revisited

Reynaldo Rojas-Hernández (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show that any $Σ_{s}$ -product of at most $𝔠$ -many $L Σ (\leq ω)$ -spaces has the $L Σ (\leq ω)$ -property. This result generalizes some known results about $L Σ (\leq ω)$ -spaces. On the other hand, we prove that every $Σ_{s}$ -product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every $Σ_{s}$ -product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

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...

On non-normality points, Tychonoff products and Suslin number

Sergei Logunov (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let a space $X$ be Tychonoff product $\prod_{α < τ} X_{α}$ of $τ$ -many Tychonoff nonsingle point spaces $X_{α}$ . Let Suslin number of $X$ be strictly less than the cofinality of $τ$ . Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification $β X$ . In particular, this is true if $X$ is either $R^{τ}$ or $ω^{τ}$ and a cardinal $τ$ is infinite and not countably cofinal.

C*-algebras have a quantitative version of Pełczyński's property (V)

Hana Krulišová (2017)

Czechoslovak Mathematical Journal

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A Banach space $X$ has Pełczyński’s property (V) if for every Banach space $Y$ every unconditionally converging operator $T : X \to Y$ is weakly compact. H. Pfitzner proved that $C^{*}$ -algebras have Pełczyński’s property (V). In the preprint (Krulišová, (2015)) the author explores possible quantifications of the property (V) and shows that $C (K)$ spaces for a compact Hausdorff space $K$ enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover,...

Pisier's inequality revisited

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 $\int_{- 1, 1 ⁿ} | | \sum_{j = 1}^{n} \partial_{j} f_{j} {(ε) | |}^{p} d μ (ε) \leq^{p} \int_{- 1, 1 ⁿ} \int_{- 1, 1 ⁿ} | | \sum_{j = 1}^{n} δ_{j} Δ f_{j} (ε) {| |}^{p} d μ (ε) d μ (δ)$ , where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and ${\partial_{j}}_{j = 1}^{n}$ and $Δ = \sum_{j = 1}^{n} \partial_{j}$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by $ⁿ_{p} (X)$ , we show that $ⁿ_{p} (X) \leq \sum_{k = 1}^{n} 1 / k$ for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...