The dual form of the approximation property for a Banach space and a subspace

T. Figiel; W. B. Johnson

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Displaying similar documents to “The dual form of the approximation property for a Banach space and a subspace”

An approximation property with respect to an operator ideal

Juan Manuel Delgado, Cándido Piñeiro (2013)

Studia Mathematica

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Given an operator ideal , we say that a Banach space X has the approximation property with respect to if T belongs to ${\bar{S \circ T : S \in ℱ (X)}}^{τ_{c}}$ for every Banach space Y and every T ∈ (Y,X), $τ_{c}$ being the topology of uniform convergence on compact sets. We present several characterizations of this type of approximation property. It is shown that some of the existing approximation properties in the literature may be included in this setting.

Projections from $L (X, Y)$ onto $K (X, Y)$

Kamil John (2000)

Commentationes Mathematicae Universitatis Carolinae

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Generalization of certain results in [Sap] and simplification of the proofs are given. We observe e.g.: Let $X$ and $Y$ be Banach spaces such that $X$ is weakly compactly generated Asplund space and $X^{*}$ has the approximation property (respectively $Y$ is weakly compactly generated Asplund space and $Y^{*}$ has the approximation property). Suppose that $L (X, Y) \neq K (X, Y)$ and let $1 < λ < 2$ . Then $X$ (respectively $Y$ ) can be equivalently renormed so that any projection $P$ of $L (X, Y)$ onto $K (X, Y)$ has the sup-norm greater or equal to $λ$ . ...

Schauder bases and the bounded approximation property in separable Banach spaces

Jorge Mujica, Daniela M. Vieira (2010)

Studia Mathematica

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Let E be a separable Banach space with the λ-bounded approximation property. We show that for each ϵ > 0 there is a Banach space F with a Schauder basis such that E is isometrically isomorphic to a 1-complemented subspace of F and, moreover, the sequence (Tₙ) of canonical projections in F has the properties $s u p_{n \in ℕ} | | T ₙ | | \leq λ + ϵ$ and $l i m s u p_{n \to \infty} | | T ₙ | | \leq λ$ . This is a sharp quantitative version of a classical result obtained independently by Pełczyński and by Johnson, Rosenthal and Zippin.

The polynomial approximation of continuous functions defined on $(- \infty, \infty)$

Leetsch C. Hsu (1959)

Czechoslovak Mathematical Journal

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Approximation of functions from $L^{p} {(ω ̃)}_{β}$ by linear operators of conjugate Fourier series

WŁodzimierz Łenski, Bogdan Szal (2011)

Banach Center Publications

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We show the results corresponding to some theorems of S. Lal and H. K. Nigam [Int. J. Math. Math. Sci. 27 (2001), 555-563] on the norm and pointwise approximation of conjugate functions and to the results of the authors [Acta Comment. Univ. Tartu. Math. 13 (2009), 11-24] also on such approximations.

The characterizations of upper approximation operators based on special coverings

Pei Wang, Qingguo Li (2017)

Open Mathematics

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In this paper, we discuss the approximation operators [...] apr¯NS ${\bar{a p r}}_{N S}$ and [...] apr¯S ${\bar{a p r}}_{S}$ which are based on NS(U) and S. We not only obtain some properties of NS(U) and S, but also give examples to show some special properties. We also study sufficient and necessary conditions when they become closure operators. In addition, we give general and topological characterizations of the covering for two types of covering-based upper approximation operators being closure operators.

Strong unicity criterion in some space of operators

Grzegorz Lewicki (1993)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a finite dimensional Banach space and let $Y \subset X$ be a hyperplane. Let $L_{Y} = {L \in L (X, Y) : L ∣_{Y} = 0}$ . In this note, we present sufficient and necessary conditions on $L_{0} \in L_{Y}$ being a strongly unique best approximation for given $L \in L (X)$ . Next we apply this characterization to the case of $X = l_{\infty}^{n}$ and to generalization of Theorem I.1.3 from [12] (see also [13]).

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.

An ideal characterization of when a subspace of certain Banach spaces has the metric compact approximation property

J. Cabello, E. Nieto (1998)

Studia Mathematica

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C.-M. Cho and W. B. Johnson showed that if a subspace E of $ℓ_{p}$ , 1 < p < ∞, has the compact approximation property, then K(E) is an M-ideal in ℒ(E). We prove that for every r,s ∈ ]0,1] with $r^{2} + s^{2} < 1$ , the James space can be provided with an equivalent norm such that an arbitrary subspace E has the metric compact approximation property iff there is a norm one projection P on ℒ(E)* with Ker P = K(E)⊥ satisfying ∥⨍∥ ≥ r∥Pf∥ + s∥φ - Pf∥ ∀⨍ ∈ ℒ(E)*. A similar result is proved for subspaces of...