scholarly journals A Gelfand-Phillips space not containing l1 whose dual ball is not weak * sequentially compact

2001 ◽  
Vol 43 (1) ◽  
pp. 125-128 ◽  
Author(s):  
Bengt Josefson
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Isomorphic Copy ◽  
Linear Functionals ◽  
Tree Space ◽  
Convergent Sequences

A set D in a Banach space E is called limited if pointwise convergent sequences of linear functionals converge uniformly on D and E is called a GP-space (after Gelfand and Phillips) if every limited set in E is relatively compact. Banach spaces with weak * sequentially compact dual balls (W*SCDB for short) are GP-spaces and l1 is a GP-space without W*SCDB. Disproving a conjecture of Rosenthal and inspired by James tree space, Hagler and Odell constructed a class of Banach spaces ([HO]-spaces) without both W*SCDB and subspaces isomorphic to l1. Schlumprecht has shown that there is a subclass of the [HO]-spaces which are also GP-spaces. It is not clear however if any [HO]-construction yields a GP-space—in fact it is not even clear that W*SCDB[lrarr ]GP-space is false in general for the class of Banach spaces containing no subspace isomorphic to l1. In this note the example of Hagler and Odell is modified to yield a GP-space without W*SCDB and without an isomorphic copy of l1.

Projections on Tree-Like banach Spaces

1985 ◽  
Vol 37 (5) ◽  
pp. 908-920
Author(s):  
A. D. Andrew
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Unconditional Basis ◽  
Reflexive Banach Space ◽  
Separable Space ◽  
Bounded Linear ◽  
Tree Space

1. In this paper, we investigate the ranges of projections on certain Banach spaces of functions defined on a diadic tree. The notion of a “tree-like” Banach space is due to James 4], who used it to construct the separable space JT which has nonseparable dual and yet does not contain l1. This idea has proved useful. In [3], Hagler constructed a hereditarily c0 tree space, HT, and Schechtman [6] constructed, for each 1 ≦ p ≦ ∞, a reflexive Banach space, STp with a 1-unconditional basis which does not contain lp yet is uniformly isomorphic to for each n.In [1] we showed that if U is a bounded linear operator on JT, then there exists a subspace W ⊂ JT, isomorphic to JT such that either U or (1 — U) acts as an isomorphism on W and UW or (1 — U)W is complemented in JT. In this paper, we establish this result for the Hagler and Schechtman tree spaces.

scholarly journals The () Property in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Danyal Soybaş
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Isomorphic Copy ◽  
Weakly Compact ◽  
Geometric Properties ◽  
Bounded Linear ◽  
Real Functions

A Banach space is said to have (D) property if every bounded linear operator is weakly compact for every Banach space whose dual does not contain an isomorphic copy of . Studying this property in connection with other geometric properties, we show that every Banach space whose dual has (V∗) property of Pełczyński (and hence every Banach space with (V) property) has (D) property. We show that the space of real functions, which are integrable with respect to a measure with values in a Banach space , has (D) property. We give some other results concerning Banach spaces with (D) property.

scholarly journals On geometric properties of ⊥BJCϵ symmetric in some types of Banach spaces

2020 ◽  
Vol 1664 (1) ◽  
pp. 012038
Author(s):  
Saied A. Jhonny ◽  
Buthainah A. A. Ahmed
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Real Banach Space ◽  
Convex Banach Space ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Geometric Properties ◽  
Strictly Convex ◽  
Strictly Convex Banach Space

Abstract In this paper, we ⊥ B J C ϵ -orthogonality and explore ⊥ B J C ϵ -symmetricity such as a ⊥ B J C ϵ -left-symmetric ( ⊥ B J C ϵ -right-symmetric) of a vector x in a real Banach space (𝕏, ‖·‖𝕩) and study the relation between a ⊥ B J C ϵ -right-symmetric ( ⊥ B J C ϵ -left-symmetric) in ℐ(x). New results and proofs are include the notion of norm attainment set of a continuous linear functionals on a reflexive and strictly convex Banach space and using these results to characterize a smoothness of a vector in a unit sphere.

Copies of l∞ in an operator space

1990 ◽  
Vol 108 (3) ◽  
pp. 523-526 ◽  
Author(s):  
Lech Drewnowski
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Operators ◽  
Operator Norm ◽  
Operator Space ◽  
Isomorphic Copy ◽  
Continuous Linear ◽  
Weakly Continuous ◽  
Continuous Linear Operators

Let X and Y be Banach spaces. Then Kw*(X*, Y) denotes the Banach space of compact and weak*-weakly continuous linear operators from X* into Y, endowed with the usual operator norm. Let us write E⊃l∞ to indicate that a Banach space E contains an isomorphic copy of l∞. The purpose of this note is to prove the followingTheorem. Kw*(X*, Y) ⊃ l∞if and only if either X ⊃ l∞or Y ⊃ l∞.

scholarly journals The compact range property and C0

1986 ◽  
Vol 28 (1) ◽  
pp. 113-114 ◽  
Author(s):  
Neil E. Gretsky ◽  
Joseph M. Ostroy
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Positive Operator ◽  
Short Note ◽  
Mathematical Economics ◽  
Compact Range ◽  
Convergent Sequences

The purpose of this short note is to make an observation about Dunford–Pettis operators from L1[0, 1] to C0. Recall that an operator T:E→F (where E and F are Banach spaces) is called Dunford–Pettis if T takes weakly convergent sequences of E into norm convergent sequences of F. A Banach space F has the Compact Range Property (CRP) if every operator T:L1]0, 1]→F is Dunford–Pettis. Talagrand shows in his book [2] that C0 does not have the CRP. It is of interest (especially in mathematical economics [3]) to note that every positive operator from L1[0, 1] to C0 is Dunford–Pettis.

scholarly journals Banach spaces of functions analytic in a polydisc

1986 ◽  
Vol 9 (1) ◽  
pp. 39-46
Author(s):  
Leon M. Hall
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Hadamard Product ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Open Unit ◽  
Linear Functionals

This paper is concerned with functions of several complex variables analytic in the unit polydisc. Certain Banach spaces to which these functions might belong are defined and some relationships between them are developed. The space of linear functionals for the Banach space of functions analytic in the open unit polydisc and continuous on the unit torus is then described in terms of analytic functions using an extension of the Hadamard product.

scholarly journals A Note on Banach Spaces Containing ℓ1

1976 ◽  
Vol 19 (3) ◽  
pp. 365-367 ◽  
Author(s):  
Robert H. Lohman
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Cauchy Sequence ◽  
Isomorphic Copy ◽  
Quotient Mapping

AbstractIf Y is a subspace of a Banach space X, either Y contains an isomorphic copy of ℓ1 or each weak Cauchy sequence in X/Y has a subsequence that is the image under the quotient mapping of a weak Cauchy sequence in X. If X is weakly sequentially complete and Y is reflexive, X/Y is weakly sequentially complete. Related structural results are given.

Conjugate Banach spaces

1957 ◽  
Vol 53 (3) ◽  
pp. 576-580 ◽  
Author(s):  
A. F. Ruston
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Complex Banach Space ◽  
Linear Mapping ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Continuous Linear Mapping ◽  
Conjugate Space ◽  
Necessary And Sufficient ◽  
Weak Topologies

The purpose of this note is to present two characterizations of conjugate Banach spaces. More precisely, we present two conditions, each necessary and sufficient for a (real or complex) Banach space to be isomorphic to the conjugate space of a Banach space, and two corresponding conditions for to be equivalent to the conjugate space of a Banach space. Other characterizations, in terms of weak topologies, have been given by Alaoglu ((1), Theorem 2:1, p. 256, and Corollary 2:1, p. 257) and Bourbaki ((4), Chap, IV, §5, exerc. 15c, p. 122). Here, by the conjugate space* of a Banach space we mean ((2), p. 188) the space of continuous linear functionals over . Two Banach spaces and are said to be isomorphic if there is a one-one continuous linear mapping of onto (its inverse is necessarily continuous by the inversion theorem ((2), Théorème 5, p. 41; (6), Theorem 2·13·7, Corollary, p. 29)); they are said to be equivalent if there is a norm-preserving linear mapping of onto .((2), p. 180).

scholarly journals Approximate fixed points for $n$-Linear functional by $(\mu,\sigma)$- nonexpansive Mappings on $n$-Banach spaces

2020 ◽  
Vol 1 (1) ◽  
pp. 20-32
Author(s):  
Basel Hardan ◽  
Jayashree Patil ◽  
Amol Bachhav ◽  
Archana Chaudhari
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Banach Spaces ◽  
Linear Functional ◽  
Nonexpansive Mappings ◽  
Bounded Subset ◽  
Linear Functionals

In this paper, we conclude that $n$-linear functionals spaces $\Im$ has approximate fixed points set, where $\Im$ is a non-empty bounded subset of an $n$-Banach space $H$ under the condition of equivalence, and we also use class of $(\mu,\sigma)$-nonexpansive mappings.

scholarly journals A dual characterisation of the existence of small combinations of slices

1988 ◽  
Vol 37 (1) ◽  
pp. 113-120 ◽  
Author(s):  
Robert Deville
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Equivalent Norm ◽  
Isomorphic Copy

We characterise, by a property of roughness, the norms of a Banach space X such that the dual unit ball has no small combination of ω*-slices. Among separable Banach spaces, the existence of an equivalent norm for this new property of roughness characterises spaces which contain an isomorphic copy of ℓ1(N).

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