Smoothability, Strong Smoothability and Dentability in Banach Spaces1

1981 ◽  
Vol 24 (1) ◽  
pp. 59-68 ◽  
Author(s):  
R. Anantharaman ◽  
T. Lewis ◽  
J. H. M. Whitfield
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Asplund Space ◽  
Differentiable Norm ◽  
Nikodym Property ◽  
Fréchet Differentiable ◽  
Conjugate Banach Space

AbstractIt is shown that dentability of the unit ball of a conjugate Banach space X* does not imply smoothability of the unit ball of X, answering a question raised by Kemp. A property called strong smoothability is introduced and is shown to be dual to dentability. The results are used to provide new proofs of the facts that X is an Asplund space whenever it has an equivalent Fréchet differentiable norm, or whenever X* has the Radon-Nikodym Property.

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 Weak convergence theorem for Passty type asymptotically nonexpansive mappings

1999 ◽  
Vol 22 (1) ◽  
pp. 217-220
Author(s):  
B. K. Sharma ◽  
B. S. Thakur ◽  
Y. J. Cho
Keyword(s):  
Banach Space ◽  
Convergence Theorem ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm ◽  
Weak Convergence Theorem ◽  
Fréchet Differentiable

In this paper, we prove a convergence theorem for Passty type asymptotically nonexpansive mappings in a uniformly convex Banach space with Fréchet-differentiable norm.

scholarly journals Weak Convergence of the Projection Type Ishikawa Iteration Scheme for Two Asymptotically Nonexpansive Nonself-Mappings

2011 ◽  
Vol 2011 ◽  
pp. 1-19
Author(s):  
Tanakit Thianwan
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Iteration Scheme ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Ishikawa Iteration ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm ◽  
Fréchet Differentiable

We study weak convergence of the projection type Ishikawa iteration scheme for two asymptotically nonexpansive nonself-mappings in a real uniformly convex Banach spaceEwhich has a Fréchet differentiable norm or its dualE*has the Kadec-Klee property. Moreover, weak convergence of projection type Ishikawa iterates of two asymptotically nonexpansive nonself-mappings without any condition on the rate of convergence associated with the two maps in a uniformly convex Banach space is established. Weak convergence theorem without making use of any of the Opial's condition, Kadec-Klee property, or Fréchet differentiable norm is proved. Some results have been obtained which generalize and unify many important known results in recent literature.

scholarly journals Each weakly countably determined Asplund space admits a Fréchet differentiable norm

1987 ◽  
Vol 36 (3) ◽  
pp. 367-374 ◽  
Author(s):  
M. Fabian
Keyword(s):  
Asplund Space ◽  
Differentiable Norm ◽  
Fréchet Differentiable ◽  
Compactly Generated

If au Asplund space is weakly countably determined, then it admits an equivalent Fréchet differentiable norm and is weakly compactly generated. If on an Asplund space there exists an equivalent Gâteaux differentiable norm, then its dual has a projectional resolution of identity.

scholarly journals Some questions about rotundity and renormings in Banach spaces

2005 ◽  
Vol 79 (1) ◽  
pp. 131-140 ◽  
Author(s):  
A. Aizpuru ◽  
F. J. Garcia-Pacheco
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Geometric Concepts ◽  
Strongly Exposed Points ◽  
Fréchet Differentiable ◽  
Rotund Banach Space

AbstractIn this paper, we show some results involving classical geometric concepts. For example, we characterize rotundity and Efimov-Stechkin property by mean of faces of the unit ball. Also, we prove that every almost locally uniformly rotund Banach space is locally uniformly rotund if its norm is Fréchet differentiable. Finally, we also provide some theorems in which we characterize the (strongly) exposed points of the unit ball using renormings.

Geometry of Spaces of Vector-Valued Harmonic Functions

1994 ◽  
Vol 46 (2) ◽  
pp. 274-283
Author(s):  
Patrick N. Dowling ◽  
Zhibao Hu ◽  
Mark A. Smith
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Unit Sphere ◽  
Harmonic Functions ◽  
Norm Convergence ◽  
Nikodym Property ◽  
Vector Valued

AbstractIt is shown that the space hp(D,X) has the Kadec-Klee property with respect to pointwise norm convergence in the Banach space X if and only if X has the Radon-Nikodym property and every point of the unit sphere of X is a denting point of the unit ball of X. In addition, it is shown that hp(D,X) is locally uniformly rotund if and only if X is locally uniformly rotund and has the Radon-Nikodym property.

New Characterizations of the Reflexivity in Terms of the Set of Norm Attaining Functionals

1998 ◽  
Vol 41 (3) ◽  
pp. 279-289 ◽  
Author(s):  
Mariá D. Acosta ◽  
Manuel Ruiz Galán
Keyword(s):  
Banach Space ◽  
Positive Result ◽  
Unit Ball ◽  
Separable Banach Space ◽  
Intersection Property ◽  
Sufficient Condition ◽  
Empty Interior ◽  
Norm Topology ◽  
Fréchet Differentiable ◽  
Norm Attaining

AbstractAs a consequence of results due to Bourgain and Stegall, on a separable Banach space whose unit ball is not dentable, the set of norm attaining functionals has empty interior (in the norm topology). First we show that any Banach space can be renormed to fail this property. Then, our main positive result can be stated as follows: if a separable Banach space X is very smooth or its bidual satisfies the w*-Mazur intersection property, then either X is reflexive or the set of norm attaining functionals has empty interior, hence the same result holds if X has the Mazur intersection property and so, if the norm of X is Fréchet differentiable. However, we prove that smoothness is not a sufficient condition for the same conclusion.

Points of Weak*-Norm Continuity in the Unit Ball of the Space WC(K; X)*

1999 ◽  
Vol 42 (1) ◽  
pp. 118-124 ◽  
Author(s):  
T. S. S. R. K. Rao
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Vector Measures ◽  
Weak Norm ◽  
Norm Continuity ◽  
Nikodym Property ◽  
Isolated Points ◽  
Dense Set

AbstractFor a compact Hausdorff space with a dense set of isolated points, we give a complete description of points of weak*-norm continuity in the dual unit ball of the space of Banach space valued functions that are continuous when the range has the weak topology. As an application we give a complete description of points of weak-norm continuity of the unit ball of the space of vector measures when the underlying Banach space has the Radon-Nikodym property.

scholarly journals Locally uniformly rotund renorming and decompositions of Banach spaces

1984 ◽  
Vol 29 (2) ◽  
pp. 259-265 ◽  
Author(s):  
V. Zizler
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dual Space ◽  
Equivalent Norm ◽  
Differentiable Norm ◽  
Fréchet Differentiable

A norm |·| of a Banach space x is called locally uniformly rotund if lim|xn−x| = 0 whenever xn, x ∈ X, and . It is shown that such an equivalent norm exists on every Banach space x which possesses a projectional resolution {pα} of the identify operator, for which all (pα+1−pα)X admit such norms. This applies, for example, for the dual space of a space with Fréchet differentiable norm.

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