scholarly journals A Banach space without a basis which has the bounded approximation property

1987 ◽  
Vol 159 (0) ◽  
pp. 81-98 ◽  
Author(s):  
Stanislaw J. Szarek
Keyword(s):  
Banach Space ◽  
Approximation Property

scholarly journals Homological characterizations of the approximation property for Banach spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 229-239 ◽  
Author(s):  
Yu. V. Selivanov
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Cohomology Group ◽  
Approximation Property ◽  
Three Dimensional ◽  
Nuclear Operators ◽  
Finite Dimensionality

Let E be a Banach space, and let N(E) be the Banach algebra of all nuclear operators on E. In this work, we shall study the homological properties of this algebra. Some of these properties turn out to be equivalent to the (Grothendieck) approximation property for E. These include:(i) biprojectivity of N(E);(ii) biflatness of N(E);(iii) homological finite-dimensionality of N(E);(iv) vanishing of the three-dimensional cohomology group, H3(N(E), N(E)).

scholarly journals Nuclear and integral polynomials

2004 ◽  
Vol 76 (2) ◽  
pp. 269-280 ◽  
Author(s):  
Raffaella Cilia ◽  
Joaquín M. Gutiérrez
Keyword(s):  
Banach Space ◽  
Approximation Property ◽  
Integral Polynomial ◽  
Nikodym Property

AbstractLet E be a Banach space whose dual E* has the approximation property, and let m be an index. We show that E* has the Radon-Nikodým property if and only if every m-homogeneous integral polynomial from E into any Banach space is nuclear. We also obtain factorization and composition results for nuclear polynomials.

scholarly journals Strongly Extreme Points and Approximation Properties

2018 ◽  
Vol 61 (3) ◽  
pp. 449-457
Author(s):  
Trond A. Abrahamsen ◽  
Petr Hájek ◽  
Olav Nygaard ◽  
Stanimir L. Troyanski
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Extreme Point ◽  
Convex Subset ◽  
Approximation Property ◽  
Extreme Points ◽  
Local Approximation ◽  
Approximation Properties ◽  
Compact Approximation ◽  
Approximation Of The Identity

AbstractWe show that if x is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at x, then x is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the suõcient conditions mentioned.

scholarly journals Compact reduction in Lipschitz-free spaces

2021 ◽  
Vol 151 (6) ◽  
pp. 1683-1699
Author(s):  
Ramón J. Aliaga ◽  
Camille Noûs ◽  
Colin Petitjean ◽  
Antonín Procházka
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
General Principle ◽  
Metric Spaces ◽  
Approximation Property ◽  
Complete Metric Space ◽  
Infinite Dimensional ◽  
Schur Property ◽  
Free Spaces ◽  
Compact Subsets

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal {F}(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal {F}(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

scholarly journals Computability-theoretic complexity of effective Banach spaces

2022 ◽  
Author(s):  
◽  
Long Qian
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Banach Spaces ◽  
Lower Bounds ◽  
Approximation Property ◽  
Upper Bounds ◽  
Space Theory ◽  
Approximation Properties ◽  
Open Problems

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

scholarly journals ALTERNATION THEOREM FOR $C(I,X)$ AND APPLICATION TO BEST LOCAL APPROXIMATION

1993 ◽  
Vol 24 (2) ◽  
pp. 135-147
Author(s):  
A. AL-ZAMEL ◽  
R. KHALIL
Keyword(s):  
Banach Space ◽  
Approximation Property ◽  
Local Approximation ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Best Local Approximation ◽  
Alternation Theorem

Let $X$ be a Banach space with the approximation property, and $C(I,X)$ the space of continuous functions defined on $I = [0,1)$ with values in $X$. Let $u_i \in C(I,X)$, $i=1,2,\cdots, n$ and $M=span\{u_1, \cdots, u_n\}$. The object of this paper is to prove that if $\{u_1, \cdots, u_n\}$ satisfies certain conditions, then for $f \in C(I,X)$ and $g \in M$ we have $||f-g|| = \inf\{||f-h|| : h\in M\}$ if and only if $f-g$ has at least $n$-zeros. An application to best local approximation in $C(I,X)$ is given.

scholarly journals Bounded approximation properties in terms of $C[0,1]$

2012 ◽  
Vol 110 (1) ◽  
pp. 45 ◽  
Author(s):  
Åsvald Lima ◽  
Vegard Lima ◽  
Eve Oja
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
Approximation Property ◽  
Integral Operators ◽  
Operator Ideal ◽  
Approximation Properties ◽  
Finite Rank Operators ◽  
Nuclear Operators ◽  
The Ideal

Let $X$ be a Banach space and let $\mathcal I$ be the Banach operator ideal of integral operators. We prove that $X$ has the $\lambda$-bounded approximation property ($\lambda$-BAP) if and only if for every operator $T\in \mathcal I(X,C[0,1]^*)$ there exists a net $(S_\alpha)$ of finite-rank operators on $X$ such that $S_\alpha\to I_X$ pointwise and 26767 \limsup_\alpha\|TS_\alpha\|_{\mathcal I}\leq\lambda\|T\|_{\mathcal I}. 26767 We also prove that replacing $\mathcal I$ by the ideal $\mathcal N$ of nuclear operators yields a condition which is equivalent to the weak $\lambda$-BAP.

THE BOUNDED APPROXIMATION PROPERTY FOR THE WEIGHTED SPACES OF HOLOMORPHIC MAPPINGS ON BANACH SPACES

2017 ◽  
Vol 60 (2) ◽  
pp. 307-320 ◽  
Author(s):  
MANJUL GUPTA ◽  
DEEPIKA BAWEJA
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Open Subset ◽  
Weighted Space ◽  
Approximation Property ◽  
Holomorphic Mappings ◽  
Countable Family ◽  
Weighted Spaces ◽  
Metric Approximation

AbstractIn this paper, we study the bounded approximation property for the weighted space$\mathcal{HV}$(U) of holomorphic mappings defined on a balanced open subsetUof a Banach spaceEand its predual$\mathcal{GV}$(U), where$\mathcal{V}$is a countable family of weights. After obtaining an$\mathcal{S}$-absolute decomposition for the space$\mathcal{GV}$(U), we show thatEhas the bounded approximation property if and only if$\mathcal{GV}$(U) has. In case$\mathcal{V}$consists of a single weightv, an analogous characterization for the metric approximation property for a Banach spaceEhas been obtained in terms of the metric approximation property for the space$\mathcal{G}_v$(U).

How bad can a Banach space with the approximation property be?

1983 ◽  
Vol 33 (6) ◽  
pp. 427-434 ◽  
Author(s):  
O. I. Reinov
Keyword(s):  
Banach Space ◽  
Approximation Property

scholarly journals A remark on a theorem of Caradus

1972 ◽  
Vol 6 (3) ◽  
pp. 355-356
Author(s):  
J.A. Johnson
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Weak Topology ◽  
Approximation Property ◽  
Approximation Problem

It is shown how a result of S.R. Caradus on the approximation problem can be obtained from known theorems.Terms used here are standard (see [1] or [3]).Let X denote a Banach space, S its unit ball in the weak topology, and X* the dual of X. In [1], the following theorem is proved: (I) If X is reflexive and X* (considered as a subspaoe of the continuous scalar-valued functions C(S) in the canonical way) is complemented in C(S), then X has the approximation property.It is our purpose to point out that (I) is a corollary to some known theorems that yield the stronger conclusion (II) below.

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