scholarly journals A weak Asplund space whose dual is not in Stegall’s class

2002 ◽  
Vol 130 (7) ◽  
pp. 2139-2143 ◽  
Author(s):  
Ondřej F. K. Kalenda
Keyword(s):  
Asplund Space ◽  
Weak Asplund Space

scholarly journals A weak Asplund space whose dual is not weak$^*$ fragmentable

2001 ◽  
Vol 129 (12) ◽  
pp. 3741-3747 ◽  
Author(s):  
Petar S. Kenderov ◽  
Warren B. Moors ◽  
Scott Sciffer
Keyword(s):  
Asplund Space ◽  
Weak Asplund Space

A Note on Minimal Usco Maps

1991 ◽  
Vol 34 (3) ◽  
pp. 412-416 ◽  
Author(s):  
Andrei Verona ◽  
Maria Elena Verona
Keyword(s):  
Banach Space ◽  
Short Proof ◽  
Lower Semicontinuous ◽  
Baire Space ◽  
Asplund Space ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Strictly Convex ◽  
Minimal Usco Map ◽  
Weak Asplund Space

AbstractWe prove that the composition of a minimal usco map, defined on a Baire space, with a lower semicontinuous function is single valued and usco at each point of a dense G$ subset of its domain. This extends earlier results of Kenderov and Fitzpatrick. As a first consequence, we prove that a Banach space, with the property that there exists a strictly convex, weak* lower semicontinuous function on its dual, is a weak Asplund space. As a second consequence, we present a short proof of the fact that a Banach space with separable dual is an Asplund space.

scholarly journals The differentiability of convex functions on topological linear spaces

1990 ◽  
Vol 42 (2) ◽  
pp. 201-213 ◽  
Author(s):  
Bernice Sharp
Keyword(s):  
Convex Functions ◽  
Inductive Limit ◽  
Asplund Space ◽  
Fréchet Spaces ◽  
Linear Spaces ◽  
Asplund Spaces ◽  
Continuous Convex Function ◽  
Mazur’S Theorem ◽  
Topological Linear ◽  
Differentiability Properties

In this paper topological linear spaces are categorised according to the differentiability properties of their continuous convex functions. Mazur's Theorem for Banach spaces is generalised: all separable Baire topological linear spaces are weak Asplund. A class of spaces is given for which Gateaux and Fréchet differentiability of a continuous convex function coincide, which with Mazur's theorem, implies that all Montel Fréchet spaces are Asplund spaces. The effect of weakening the topology of a given space is studied in terms of the space's classification. Any topological linear space with its weak topology is an Asplund space; at the opposite end of the topological spectrum, an example is given of the inductive limit of Asplund spaces which is not even a Gateaux differentiability space.

scholarly journals On a Certain Generalized Functional Equation for Set-Valued Functions

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2243
Author(s):  
Yaroslav Bazaykin ◽  
Dušan Bednařík ◽  
Veronika Borůvková ◽  
Tomáš Zuščák
Keyword(s):  
Functional Equation ◽  
Integral Functional ◽  
Asplund Space ◽  
Invariant Metric ◽  
Real Vector ◽  
Linear Metric Space ◽  
Real Linear ◽  
The Right ◽  
Locally Convex ◽  
Generalized Type

The aim of the paper is to generalize results by Sikorska on some functional equations for set-valued functions. In the paper, a tool is described for solving a generalized type of an integral-functional equation for a set-valued function F:X→cc(Y), where X is a real vector space and Y is a locally convex real linear metric space with an invariant metric. Most general results are described in the case of a compact topological group G equipped with the right-invariant Haar measure acting on X. Further results are found if the group G is finite or Y is Asplund space. The main results are applied to an example where X=R2 and Y=Rn, n∈N, and G is the unitary group U(1).

scholarly journals Balls intersection properties of Banach spaces

1992 ◽  
Vol 45 (2) ◽  
pp. 333-342 ◽  
Author(s):  
Dongjian Chen ◽  
Zhibao Hu ◽  
Bor-Luh Lin
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Intersection Property ◽  
Necessary And Sufficient Conditions ◽  
Asplund Space ◽  
Necessary And Sufficient

Necessary and sufficient conditions for a Banach space with the Mazur intersection property to be an Asplund space are given. It is proved that Mazur intersection property is determined by the separable subspaces of the space. Corresponding problems for a space to have the ball-generated property are considered. Some comments on possible renorming so that a space having the Mazur intersection property are given.

scholarly journals Smoothness and the asymptotic-norming properties of Banach spaces

1992 ◽  
Vol 45 (2) ◽  
pp. 285-296 ◽  
Author(s):  
Zhibao Hu ◽  
Bor-Luh Lin
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dual Space ◽  
Asplund Space ◽  
Duality Mapping

We study some smoothness properties of a Banach space X that are related to the weak* asymptotic-norming properties of the dual space X*. These properties imply that X is an Asplund space and are related to the duality mapping of X.

scholarly journals A CONTINUITY CHARACTERIZATION OF ASPLUND SPACES

2011 ◽  
Vol 83 (3) ◽  
pp. 450-455
Author(s):  
J. R. GILES
Keyword(s):  
Banach Space ◽  
Asplund Space ◽  
Weakly Compact ◽  
Asplund Spaces ◽  
Subdifferential Mapping

AbstractA Banach space is an Asplund space if every continuous gauge has a point where the subdifferential mapping is Hausdorff weak upper semi-continuous with weakly compact image. This contributes towards the solution of a problem posed by Godefroy, Montesinos and Zizler.

scholarly journals On the characterisation of Asplund spaces

1982 ◽  
Vol 32 (1) ◽  
pp. 134-144 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Direct Proof ◽  
Asplund Space ◽  
Open Convex ◽  
Asplund Spaces ◽  
Separable Subspace ◽  
Nikodym Property ◽  
Continuous Convex Function ◽  
Fréchet Differentiable ◽  
Open Convex Subset

AbstractA Banach space is an Asplund space if every continuous convex function on an open convex subset is Fréchet differentiable on a dense G8 subset of its domain. The recent research on the Radon-Nikodým property in Banach spaces has revealed that a Banach space is an Asplund space if and only if every separable subspace has separable dual. It would appear that there is a case for providing a more direct proof of this characterisation.

scholarly journals FRÉCHET INTERMEDIATE DIFFERENTIABILITY OF LIPSCHITZ FUNCTIONS ON ASPLUND SPACES

2009 ◽  
Vol 79 (2) ◽  
pp. 309-317 ◽  
Author(s):  
J. R. GILES
Keyword(s):  
Large Class ◽  
Open Subset ◽  
Lipschitz Function ◽  
Dense Subset ◽  
Lipschitz Functions ◽  
Nonempty Open Subset ◽  
Asplund Space ◽  
Asplund Spaces ◽  
Fréchet Differentiable

AbstractThe deep Preiss theorem states that a Lipschitz function on a nonempty open subset of an Asplund space is densely Fréchet differentiable. However, the simpler Fabian–Preiss lemma implies that it is Fréchet intermediately differentiable on a dense subset and that for a large class of Lipschitz functions this dense subset is residual. Results are presented for Asplund generated spaces.

scholarly journals Asymptotic-norming and Mazur intersection properties in Bochner function spaces

1993 ◽  
Vol 48 (2) ◽  
pp. 177-186 ◽  
Author(s):  
Zhibao Hu ◽  
Bor-Luh Lin
Keyword(s):  
Banach Space ◽  
Function Space ◽  
Function Spaces ◽  
Intersection Property ◽  
Asplund Space

A Banach space X has the asymptotic-norming property if and only if the Lebesgue-Bochner function space Lp (μ, X) has the asumptotic-norming property for p with 1 < p < ∞. It follows that a Banach space X is Hahn-Banach smooth if and only if Lp (μ, X) is Hahn-Banach smooth for p with 1 < p < ∞. We also show that for p with 1 < p < ∞, (1) if X has the compact Mazur intersection property then so does Lp(μ, X); (2) if the measure μ is not purely atomic, then the space Lp(μ, X) has the Mazur intersection property if and only if X is an Asplund space and has the Mazur intersection property.

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