On the classes of hereditarily $\ell _p$ Banach spaces

Parviz Azimi; A. A. Ledari

Displaying similar documents to “On the classes of hereditarily $ℓ_{p}$ Banach spaces”

On the number of non-isomorphic subspaces of a Banach space

Valentin Ferenczi, Christian Rosendal (2005)

Studia Mathematica

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We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following. Let be a Banach space with an unconditional basis ${(e_{i})}_{i \in ℕ}$ ; then either there exists a perfect set P of infinite subsets of ℕ such that for any two distinct A,B ∈ P, ${[e_{i}]}_{i \in A} ≇ {[e_{i}]}_{i \in B}$ , or for a residual set of infinite subsets A of ℕ, ${[e_{i}]}_{i \in A}$ is isomorphic to , and in that case, is isomorphic to its square, to its hyperplanes, uniformly isomorphic to $\oplus {[e_{i}]}_{i \in D}$ for any D ⊂ ℕ, and isomorphic to a denumerable Schauder...

The Dual of a Non-reflexive L-embedded Banach Space Contains $l^{\infty}$ Isometrically

Hermann Pfitzner (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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A Banach space is said to be L-embedded if it is complemented in its bidual in such a way that the norm between the two complementary subspaces is additive. We prove that the dual of a non-reflexive L-embedded Banach space contains $l^{\infty}$ isometrically.

Estimation of the Szlenk index of Banach spaces via Schreier spaces

Ryan Causey (2013)

Studia Mathematica

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For each ordinal α < ω₁, we prove the existence of a Banach space with a basis and Szlenk index $ω^{α + 1}$ which is universal for the class of separable Banach spaces with Szlenk index not exceeding $ω^{α}$ . Our proof involves developing a characterization of which Banach spaces embed into spaces with an FDD with upper Schreier space estimates.

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.

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.

On the mutually non isomorphic $l_{p} (l_{q})$

Pilar Cembranos, Jose Mendoza (2016)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this note we survey the partial results needed to show the following general theorem: $l_{p} (l_{q}) : 1 \leq p, q \leq + \infty$ is a family of mutually non isomorphic Banach spaces. We also comment some related facts and open problems.

An indecomposable and unconditionally saturated Banach space

Spiros A. Argyros, Antonis Manoussakis (2003)

Studia Mathematica

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We construct an indecomposable reflexive Banach space $X_{i u s}$ such that every infinite-dimensional closed subspace contains an unconditional basic sequence. We also show that every operator $T \in ℬ (X_{i u s})$ is of the form λI + S with S a strictly singular operator.

Universality, complexity and asymptotically uniformly smooth Banach spaces

Ryan M. Causey, Gilles Lancien (2023)

Commentationes Mathematicae Universitatis Carolinae

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For $1 < p \leq \infty$ , we show the existence of a Banach space which is both injectively and surjectively universal for the class of all separable Banach spaces with an equivalent $p$ -asymptotically uniformly smooth norm. We prove that this class is analytic complete in the class of separable Banach spaces. These results extend previous works by N. J. Kalton, D. Werner and O. Kurka in the case $p = \infty$ .

On the Banach-Mazur distance between continuous function spaces with scattered boundaries

Jakub Rondoš (2023)

Czechoslovak Mathematical Journal

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We study the dependence of the Banach-Mazur distance between two subspaces of vector-valued continuous functions on the scattered structure of their boundaries. In the spirit of a result of Y. Gordon (1970), we show that the constant $2$ appearing in the Amir-Cambern theorem may be replaced by $3$ for some class of subspaces. We achieve this by showing that the Banach-Mazur distance of two function spaces is at least 3, if the height of the set of weak peak points of one of the spaces differs...

Spaces of operators and c₀

P. Lewis (2001)

Studia Mathematica

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Bessaga and Pełczyński showed that if c₀ embeds in the dual X* of a Banach space X, then ℓ¹ embeds complementably in X, and $ℓ^{\infty}$ embeds as a subspace of X*. In this note the Diestel-Faires theorem and techniques of Kalton are used to show that if X is an infinite-dimensional Banach space, Y is an arbitrary Banach space, and c₀ embeds in L(X,Y), then $ℓ^{\infty}$ embeds in L(X,Y), and ℓ¹ embeds complementably in $X \otimes_{γ} Y *$ . Applications to embeddings of c₀ in various spaces of operators are given.

On monotonic functions from the unit interval into a Banach space with uncountable sets of points of discontinuity

Artur Michalak (2003)

Studia Mathematica

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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. We show that if f: [0,1] → X is an increasing function with respect to a norming subset E of X* with uncountably many points of discontinuity and Q is a countable dense subset of [0,1], then (1) $\bar{l i n f ([0, 1])}$ contains an order isomorphic copy of D(0,1), (2) $\bar{l i n f (Q)}$ contains an isomorphic copy of C([0,1]), (3) $\bar{l i n f ([0, 1])} / \bar{l i n f (Q)}$ contains an isomorphic copy of c₀(Γ) for some uncountable...

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...

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.

On complemented copies of c₀(ω₁) in C(Kⁿ) spaces

Leandro Candido, Piotr Koszmider (2016)

Studia Mathematica

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Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(Kⁿ) or equivalently the n-fold injective tensor product $\otimes ̂_{ε}^{n} C (K)$ or the Banach space of vector valued continuous functions C(K,C(K,C(K...,C(K)...). We address the question of the existence of complemented copies of c₀(ω₁) in $\otimes ̂_{ε}^{n} C (K)$ under the hypothesis that C(K) contains such a copy. This is related to the results of E. Saab and P. Saab that $X \otimes ̂_{ε} Y$ contains a complemented copy of c₀ if one of the infinite-dimensional...

On the existence of non-linear frames

Shah Jahan, Varinder Kumar, S.K. Kaushik (2017)

Archivum Mathematicum

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A stronger version of the notion of frame in Banach space called Strong Retro Banach frame (SRBF) is defined and studied. It has been proved that if $𝒳$ is a Banach space such that $𝒳^{*}$ has a SRBF, then $𝒳$ has a Bi-Banach frame with some geometric property. Also, it has been proved that if a Banach space $𝒳$ has an approximative Schauder frame, then $𝒳^{*}$ has a SRBF. Finally, the existence of a non-linear SRBF in the conjugate of a separable Banach space has been proved.

On pairs of Banach spaces which are isomorphic to complemented subspaces of each other

Elói Medina Galego (2004)

Colloquium Mathematicae

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We establish the existence of Banach spaces E and F isomorphic to complemented subspaces of each other but with $E^{m} \oplus F ⁿ$ isomorphic to $E^{p} \oplus F^{q}$ , m, n, p, q ∈ ℕ, if and only if m = p and n = q.

Banach spaces of bounded Szlenk index II

D. Freeman, E. Odell, Th. Schlumprecht, A. Zsák (2009)

Fundamenta Mathematicae

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For every α < ω₁ we establish the existence of a separable Banach space whose Szlenk index is $ω^{α ω + 1}$ and which is universal for all separable Banach spaces whose Szlenk index does not exceed $ω^{α ω}$ . In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with Tsirelson type upper estimates.

Estimation of the Szlenk index of reflexive Banach spaces using generalized Baernstein spaces

Ryan Causey (2015)

Fundamenta Mathematicae

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For each ordinal α < ω₁, we prove the existence of a separable, reflexive Banach space W with a basis so that Sz(W), $S z (W *) \leq ω^{α + 1}$ which is universal for the class of separable, reflexive Banach spaces X satisfying Sz(X), $S z (X *) \leq ω^{α}$ .

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.

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...

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...

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.

(Non-)amenability of ℬ(E)

Volker Runde (2010)

Banach Center Publications

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In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra ℬ(E) of all bounded linear operators on a Banach space E could ever be amenable if dim E = ∞. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros-Haydon result that solves the “scalar plus compact problem”: there is an infinite-dimensional Banach space E, the dual of which is ℓ¹, such that $ℬ (E) = (E) + ℂ i d_{E}$ . Still, ℬ(ℓ²) is...

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.

Geometry of the Banach spaces C(βℕ × K,X) for compact metric spaces K

Dale E. Alspach, Elói Medina Galego (2011)

Studia Mathematica

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A classical result of Cembranos and Freniche states that the C(K,X) space contains a complemented copy of c₀ whenever K is an infinite compact Hausdorff space and X is an infinite-dimensional Banach space. This paper takes this result as a starting point and begins a study of conditions under which the spaces C(α), α < ω₁, are quotients of or complemented in C(K,X). In contrast to the c₀ result, we prove that if C(βℕ ×[1,ω],X) contains a complemented copy of $C (ω^{ω})$ then X contains a copy...

Embeddings of C(K) spaces into C(S,X) spaces with distortion strictly less than 3

Leandro Candido, Elói Medina Galego (2013)

Fundamenta Mathematicae

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In the spirit of the classical Banach-Stone theorem, we prove that if K and S are intervals of ordinals and X is a Banach space having non-trivial cotype, then the existence of an isomorphism T from C(K, X) onto C(S,X) with distortion $| | T | | | | T^{- 1} | |$ strictly less than 3 implies that some finite topological sum of K is homeomorphic to some finite topological sum of S. Moreover, if Xⁿ contains no subspace isomorphic to $X^{n + 1}$ for every n ∈ ℕ, then K is homeomorphic to S. In other words, we obtain a vector-valued...

Displaying similar documents to “On the classes of hereditarily ℓ p Banach spaces”

Displaying similar documents to “On the classes of hereditarily $ℓ_{p}$ Banach spaces”