On Banach Spaces Without the Approximation Property †

2016

We prove that the kernel of a quotient operator from an L 1-space onto a Banach space X with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky-case ℓ 1-and Figiel, Johnson and Pe lczyński-case X * separable. Given a Banach space X, we show that if the kernel of a quotient map from some L 1-space onto X has the BAP then every kernel of every quotient map from any L 1-space onto X has the BAP. The dual result for L∞-spaces also hold: if for some L∞-space E some quotient E/X has the BAP then for every L∞-space E every quotient E/X has the BAP.

Israel Journal of Mathematics, 2013

We prove that the kernel of a quotient operator from an L 1 -space onto a Banach space X with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky -case ℓ 1 -and Figiel, Johnson and Pe lczyński -case X * separable. Given a Banach space X, we show that if the kernel of a quotient map from some L 1 -space onto X has the BAP then every kernel of every quotient map from any L 1 -space onto X has the BAP. The dual result for L∞-spaces also hold: if for some L∞-space E some quotient E/X has the BAP then for every L∞-space E every quotient E/X has the BAP.

It is known that any separable Banach space with BAP is a complemented subspace of a Banach space with a basis. We show that every operator with bounded approximation property, acting from a separable Banach space, can be factored through a Banach space with a basis.

Journal of Convex Analysis

We give a counterexample to a recent statement in the metric approximation theory and provide a setting where the statement holds. Let (X, •) be a real Banach space. Our notation is standard. We follow, for example, [FHHMPZ01]. In this note, if no reference to a different topology on X is made, convergence in X means •-convergence. The following concepts in the geometry of Banach spaces are more or less standard. A non-empty subset C of X is said to be approximately compact if for every x ∈ X and every sequence (c n) in C such that x − c n → dist (x, C) (such a sequence is called an approximate sequence for x in C), then (c n) has a convergent subsequence. The set C is said to be proximinal if, for every x ∈ X, the set P C (x) := {c ∈ C; x − c = dist (x, C)} is non-empty (the multivalued mapping P C : X → 2 X is called the metric projection onto C). The set C is said to be semi-Chebyshev if P C (x) contains at most one point for every x ∈ X. The set C is said to be Chebyshev if it is simultaneously proximinal and semi-Chebyshev. In this case we put P C (x) = {π C (x)} for all x ∈ X. A Banach space X is said to be locally uniformly rotund (LUR, for short) if for every x ∈ S X and every sequence (x n) in S X such that x + x n → 2, then x n → x. A Banach space X is said to be midpoint locally uniformly rotund (MLUR, for short) if for any x 0 , x n and y n in S X , n ∈ N, such that x n + y n − 2x 0 → 0, then x n − y n → 0. Every LUR Banach space is MLUR. Recall, too, that a Banach space X has property (H) (sometimes also called Kadec-Klee property) if every sequence in S X that w-converges to a point x in S X converges (to x). As it is well known, every LUR space has property (H). A Banach space X is rotund (also called strictly convex) if every point x ∈ S X is extremal. Obviously, every MLUR space is rotund. Let C be a non-empty subset of a Banach space X and let x * ∈ S X * bounded above on C. We denote S(C, x * , δ) the δ-section defined by x * in C, i.e., S(C, x * , δ) := {x ∈ C; x, x * ≥ sup C x * − δ}.

2016

Mathematische Nachrichten, 1985

It is unknown whether the HARDY space Hhas the approximation property. However, it will be shown that for each p f 1 Ha has the approximation property AP,, defined below (see also [6]), and, moreover, Hn has the approximation property ''up to log n" (see Theorem 9). Definition.-4 BANACH space , Y has the property-4P, if the canonical mapping

Bulletin of The Iranian Mathematical Society, 2009

The purpose of this paper is to introduce and to dis- cuss the concept of p-approximation and p-orthogonality in vector spaces, and to obtain some results on p- orthogonality in vector spaces similar to some well known results on the orthogonality in normed spaces. We also discuss the concept of p-extension of linear functionals on a vector space, and give a characterization of linear functionals on a subspace having a unique p-extension Hahn-Banach to the whole vector space.

2016

Abstract. A subspace Y of a Banach space X is an almost constrained (AC) subspace if any family of closed balls centred at points of Y that intersects in X also intersects in Y. In this paper, we show that a subspace H of finite codimension in C(K), the space of continuous functions on a compact Hausdorff space K, is an AC-subspace if and only if H is the range of a norm one projection in C(K). We also give a simple proof that the implication “AC ⇒ 1-complemented ” holds for any subspace of c0(Γ) and c. 1.

Studia Mathematica, 2020

For a fixed Banach operator ideal A, we characterize A-compact sets (in the sense of Carl and Stephani) that are determined by c0 via the Banach composition ideal A • K∞, with K∞ the Banach ideal of Fourie and Swart. This characterization allows us to relate KA-approximation properties on a Banach space and KB-approximation properties on its dual space, where A and B are ideals linked by some classical procedures. These approximation properties have been widely studied in several papers in the last years.

2010

It is a translation of an old paper of mine. We describe the topology tau_p in the space Pi_p(Y,X), for which the closures of convex sets in tau_p and in *-weak topology of the space Pi_p(Y,X) are coincident. Thereafter, we investigate some properties of the space Pi_p, related to this new topology. 2010-remark: Occasionally, the topology is coincides with the lambda_p-topology from the paper "Compact operators which factor through subspaces of l_p", Math. Nachr. 281(2008), 412-423 by Deba Prasad Sinha and Anil Kumar Karn.