Lattice Banach spaces, order-isomorphic to l1

1983 ◽  
Vol 94 (3) ◽  
pp. 519-522 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Positive Cone ◽  
Schauder Basis ◽  
Ordered Banach Space

It is known, (see [7], theorem 4 or [8], corollary II, 10.1), that a positive generated, ordered Banach space X is order-isomorphic to l1 iff X has a Schauder basis which generates its positive cone X+ and X+ has a bounded base.

scholarly journals Positive coincidence points for a class of nonlinear operators and their applications to matrix equations

2020 ◽  
Vol 18 (1) ◽  
pp. 858-872
Author(s):  
Imed Kedim ◽  
Maher Berzig ◽  
Ahdi Noomen Ajmi
Keyword(s):  
Banach Space ◽  
Positive Solution ◽  
Positive Definite ◽  
Positive Cone ◽  
Matrix Equations ◽  
Ordered Banach Space ◽  
Coincidence Points ◽  
Nonlinear Operators ◽  
Nonlinear Matrix

Abstract Consider an ordered Banach space and f,g two self-operators defined on the interior of its positive cone. In this article, we prove that the equation f(X)=g(X) has a positive solution, whenever f is strictly \alpha -concave g-monotone or strictly (-\alpha ) -convex g-antitone with g super-homogeneous and surjective. As applications, we show the existence of positive definite solutions to new classes of nonlinear matrix equations.

scholarly journals Monotonic norms in ordered Banach spaces

1988 ◽  
Vol 45 (2) ◽  
pp. 217-219 ◽  
Author(s):  
K. F. Ng ◽  
C. K. Law
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dual Space ◽  
Alternative Proof ◽  
Ordered Banach Space ◽  
Dual Result ◽  
Ordered Banach Spaces

AbstractLet B be an ordered Banach space with ordered Banach dual space. Let N denote the canonical half-norm. We give an alternative proof of the following theorem of Robinson and Yamamuro: the norm on B is α-monotone (α ≥ 1) if and only if for each f in B* there exists g ∈ B* with g ≥ 0, f and ∥g∥ ≤ α N(f). We also establish a dual result characterizing α-monotonicity of B*.

scholarly journals Strongly exposed points in bases for the positive cone of ordered Banach spaces and characterizations of l1(Г)

1986 ◽  
Vol 29 (2) ◽  
pp. 271-282 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Sets ◽  
Extreme Points ◽  
Positive Cone ◽  
Infinite Dimensional ◽  
Strongly Exposed Points ◽  
Nikodym Property ◽  
Choquet Theory ◽  
Ordered Banach Spaces

The study of extreme, strongly exposed points of closed, convex and bounded sets in Banach spaces has been developed especially by the interconnection of the Radon–Nikodým property with the geometry of closed, convex and bounded subsets of Banach spaces [5],[2] . In the theory of ordered Banach spaces as well as in the Choquet theory, [4], we are interested in the study of a special type of convex sets, not necessarily bounded, namely the bases for the positive cone. In [7] the geometry (extreme points, dentability) of closed and convex subsets K of a Banach space X with the Radon-Nikodým property is studied and special emphasis has been given to the case where K is a base for acone P of X. In [6, Theorem 1], it is proved that an infinite-dimensional, separable, locally solid lattice Banach space is order-isomorphic to l1 if, and only if, X has the Krein–Milman property and its positive cone has a bounded base.

On the Construction of Sequence Spaces that have Schauder Bases

1966 ◽  
Vol 18 ◽  
pp. 1281-1293 ◽  
Author(s):  
William Ruckle
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sequence Spaces ◽  
Schauder Basis ◽  
Close Connection ◽  
Original Source ◽  
Schauder Bases ◽  
Bk Spaces ◽  
Made In

It is known that every Banach space which possesses a Schauder basis is essentially a space of sequences (6, Section 11.4). The primary objectives of this paper are: (1) to illustrate the close connection between sectionally bounded BK spaces and Banach spaces which have a Schauder basis, and (2) to consider some results in these theories in such a way as to render them easy and natural. In order to reach the largest number of readers we shall use (6) as the sole basis of our discussion. References to other authors are made in order to direct the reader to the original source of a theorem or to a related discussion.

scholarly journals Common fixed point theorems for a pair of countably condensing mappings in ordered banach spaces

2003 ◽  
Vol 16 (3) ◽  
pp. 243-248 ◽  
Author(s):  
B. C. Dhage ◽  
Donal O'Regan ◽  
Ravi P. Agarwal
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Ordered Banach Space ◽  
Isotone Mappings ◽  
Common Fixed Point Theorems ◽  
Ordered Banach Spaces

In this paper some common fixed point theorems for a pair of multivalued weakly isotone mappings on an ordered Banach space are proved.

scholarly journals Symmetry groups on ordered Banach spaces

1986 ◽  
Vol 33 (2) ◽  
pp. 177-187 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Symmetry Groups ◽  
Ordered Banach Space ◽  
Group Of Symmetries ◽  
Ordered Banach Spaces

A symmetry of an ordered Banach space is an order and norm isomorphism which commutes with its ideal centre. A class of ordered Banach spaces is introduced to show that, for a space in this class, the group of symmetries is trivial if and only if the space is lattice-ordered. When this group becomes larger, the space approaches an antilattice. This phenomenon is also investigated.

On the Classification of Biorthogonal Sequences

1974 ◽  
Vol 26 (3) ◽  
pp. 721-733 ◽  
Author(s):  
William H. Ruckle
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Separable Banach Space ◽  
Continuous Functions ◽  
Schauder Basis

The work of various authors (e.g. Frink [3] and Markushevitch [7]) suggests the possibility of studying complete biorthogonal sequences in Banach spaces as a generalization of orthogonal families of continuous functions. But except for the case where the complete biorthogonal sequence is a Schauder basis such studies have not led to a very rich theory. The main reason for this is that an arbitrary complete biorthogonal sequence is not likely to have many helpful properties. For instance, in every separable Banach space X one can find a complete biorthogonal sequence {ei, Ei} which is not one-summable.

scholarly journals Fixed Point Theorems in Ordered Banach Spaces and Applications to Nonlinear Integral Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Nawab Hussain ◽  
Mohamed-Aziz Taoudi
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Ordered Banach Space ◽  
Nonlinear Integral Equations ◽  
Common Fixed Point Theorems ◽  
Ordered Banach Spaces ◽  
Nonlinear Mappings

We present some new common fixed point theorems for a pair of nonlinear mappings defined on an ordered Banach space. Our results extend several earlier works. An application is given to show the usefulness and the applicability of the obtained results.

scholarly journals Uniformly factoring weakly compact operators and parametrised dualisation

2021 ◽  
Vol 9 ◽  
Author(s):  
L. Antunes ◽  
K. Beanland ◽  
B. M. Braga
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Borel Subset ◽  
Compact Operators ◽  
Schauder Basis ◽  
Bounded Operators ◽  
Weakly Compact ◽  
Weakly Compact Operator ◽  
Borel Map ◽  
Weakly Compact Operators

Abstract This article deals with the problem of when, given a collection $\mathcal {C}$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space Z with a Schauder basis so that every element in $\mathcal {C}$ factors through Z (or through a subspace of Z). In particular, we show that there exists a reflexive space Z with a Schauder basis so that for each separable Banach space X, each weakly compact operator from X to $L_1[0,1]$ factors through Z. We also prove the following descriptive set theoretical result: Let $\mathcal {L}$ be the standard Borel space of bounded operators between separable Banach spaces. We show that if $\mathcal {B}$ is a Borel subset of weakly compact operators between Banach spaces with separable duals, then for $A \in \mathcal {B}$ , the assignment $A \to A^*$ can be realised by a Borel map $\mathcal {B}\to \mathcal {L}$ .

scholarly journals Pre-Schauder Bases in Topological Vector Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1026 ◽  
Author(s):  
Francisco Javier García-Pacheco ◽  
Francisco Javier Pérez-Fernández
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Banach Spaces ◽  
Topological Vector Space ◽  
Linear Span ◽  
Vector Spaces ◽  
Schauder Basis ◽  
Topological Vector Spaces ◽  
Hausdorff Topological Vector Space ◽  
Schauder Bases

A Schauder basis in a real or complex Banach space X is a sequence ( e n ) n ∈ N in X such that for every x ∈ X there exists a unique sequence of scalars ( λ n ) n ∈ N satisfying that x = ∑ n = 1 ∞ λ n e n . Schauder bases were first introduced in the setting of real or complex Banach spaces but they have been transported to the scope of real or complex Hausdorff locally convex topological vector spaces. In this manuscript, we extend them to the setting of topological vector spaces over an absolutely valued division ring by redefining them as pre-Schauder bases. We first prove that, if a topological vector space admits a pre-Schauder basis, then the linear span of the basis is Hausdorff and the series linear span of the basis minus the linear span contains the intersection of all neighborhoods of 0. As a consequence, we conclude that the coefficient functionals are continuous if and only if the canonical projections are also continuous (this is a trivial fact in normed spaces but not in topological vector spaces). We also prove that, if a Hausdorff topological vector space admits a pre-Schauder basis and is w * -strongly torsionless, then the biorthogonal system formed by the basis and its coefficient functionals is total. Finally, we focus on Schauder bases on Banach spaces proving that every Banach space with a normalized Schauder basis admits an equivalent norm closer to the original norm than the typical bimonotone renorming and that still makes the basis binormalized and monotone. We also construct an increasing family of left-comparable norms making the normalized Schauder basis binormalized and show that the limit of this family is a right-comparable norm that also makes the normalized Schauder basis binormalized.

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