scholarly journals A selection theorem for weak upper semi-continuous set-valued mappings

1996 ◽  
Vol 53 (2) ◽  
pp. 213-227 ◽  
Author(s):  
Warren B. Moors
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Weak Topology ◽  
Weak Compactness ◽  
Baire Space ◽  
Selection Theorem ◽  
Set Valued Mapping ◽  
Set Valued Mappings

Let Φ be a set-valued mapping from a Baire space T into non-empty closed subsets of a Banach space X, which is upper semi-continuous with respect to the weak topology on X. In this paper, we give a condition on T which is sufficient to ensure that Φ admits a selection which is norm continuous at each point of a dense and Gδ subset of T. We also derive a variation of James' characterisation of weak compactness, which we use in conjunction with our selection theorem, to deduce some differentiability results for continuous convex functions defined on dual Banach spaces.

scholarly journals The implications for differentiability of a weak index of non-compactness

1993 ◽  
Vol 48 (1) ◽  
pp. 75-91 ◽  
Author(s):  
John R. Giles ◽  
Warren B. Moors
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Baire Space ◽  
Continuity Property ◽  
Set Valued Mappings

In a recent paper the authors showed that certain set-valued mappings from a Baire space into subsets of a Banach space which have a continuity property defined in terms of Kuratowski's index of non-compactness have inherent single-valued properties. Here we generalise the continuity property to one defined in terms of a weak index of non-compactness and we show that this wider class of set-valued mappings also has significant implications for the differentiability of convex functions on Banach spaces.

scholarly journals Generic continuity of minimal set-valued mappings

1997 ◽  
Vol 63 (2) ◽  
pp. 238-262 ◽  
Author(s):  
W. B. Moors ◽  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Metric Space ◽  
Complete Metric Space ◽  
Minimal Set ◽  
Set Valued Mapping ◽  
Set Valued Mappings ◽  
Permanence Properties ◽  
Generic Continuity ◽  
Sufficiency Conditions

AbstractWe study classes of Banach spaces where every set-valued mapping from a complete metric space into subsets of the Banach space which satisfies certain minimal properties, is single-valued and norm upper semi-continuous at the points of a dense Gδ subset of its domain. Characterisations of these classes are developed and permanence properties are established. Sufficiency conditions for membership of these classes are defined in terms of fragmentability and σ-fragmentability of the weak topology. A characterisation of non membership is used to show that l∞ (N) is not a member of our classe of generic continuity spaces.

Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions

1999 ◽  
Vol 42 (2) ◽  
pp. 139-148 ◽  
Author(s):  
José Bonet ◽  
Paweł Dománski ◽  
Mikael Lindström
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Essential Norm ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Point Evaluation ◽  
Weighted Banach Spaces ◽  
Spaces Of Analytic Functions

AbstractEvery weakly compact composition operator between weighted Banach spaces of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

scholarly journals On weak Hadamard differentiability of convex functions on Banach spaces

1996 ◽  
Vol 54 (1) ◽  
pp. 155-166 ◽  
Author(s):  
J.R. Giles ◽  
Scott Sciffer
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Directional Differentiability ◽  
Hadamard Differentiability ◽  
Hadamard Directional Differentiability

We study two variants of weak Hadamard differentiability of continuous convex functions on a Banach space, uniform weak Hadamard differentiability and weak Hadamard directional differentiability, and determine their special properties on Banach spaces which do not contain a subspace topologically isomorphic to l1.

scholarly journals NEAREST POINTS AND DELTA CONVEX FUNCTIONS IN BANACH SPACES

2015 ◽  
Vol 93 (2) ◽  
pp. 283-294
Author(s):  
JONATHAN M. BORWEIN ◽  
OHAD GILADI
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Closed Set ◽  
Nearest Point ◽  
Set Of Points ◽  
Nearest Points

Given a closed set$C$in a Banach space$(X,\Vert \cdot \Vert )$, a point$x\in X$is said to have a nearest point in$C$if there exists$z\in C$such that$d_{C}(x)=\Vert x-z\Vert$, where$d_{C}$is the distance of$x$from$C$. We survey the problem of studying the size of the set of points in$X$which have nearest points in$C$. We then turn to the topic of delta convex functions and indicate how it is related to finding nearest points.

Smoothness, Convexity, Porosity, and Separable Determination

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Equivalent Norm ◽  
Fréchet Differentiability ◽  
Smooth Norm ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Set Of Points

This chapter shows how spaces with separable dual admit a Fréchet smooth norm. It first considers a criterion of the differentiability of continuous convex functions on Banach spaces before discussing Fréchet smooth and nonsmooth renormings and Fréchet differentiability of convex functions. It then describes the connection between porous sets and Fréchet differentiability, along with the set of points of Fréchet differentiability of maps between Banach spaces. It also examines the concept of separable determination, the relevance of the σ‎-porous sets for differentiability and proves the existence of a Fréchet smooth equivalent norm on a Banach space with separable dual. The chapter concludes by explaining how one can show that many differentiability type results hold in nonseparable spaces provided they hold in separable ones.

Uniform Kadec–Klee–Huff properties of vector-valued Hardy spaces

1993 ◽  
Vol 114 (1) ◽  
pp. 25-30 ◽  
Author(s):  
P. N. Dowling ◽  
C. J. Lennard
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hardy Spaces ◽  
Weak Topology ◽  
Uniformly Convex ◽  
Vector Valued

AbstractIn [8] Partington showed that a Banach space X is uniformly convex if and only if Lp([0, 1], X) has the uniform Kadec–Klee–Huff property with respect to the weak topology (UKKH (weak)), where 1 < p < ∞. In this note we will characterize the Banach spaces X such that HP(D, X) has UKKH (weak), where 1 ≤ p < ∞. Similar results for UKKH (weak*) are also obtained. These results (and proofs) are quite different from Partington's result (and proof).

scholarly journals The fixed-point property in Banach spaces containing a copy ofc0

2003 ◽  
Vol 2003 (3) ◽  
pp. 183-192
Author(s):  
Maria A. Japón Pineda
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Banach Spaces ◽  
Weak Topology ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
Asymptotically Nonexpansive ◽  
Locally Convex Topology ◽  
Locally Convex

We prove that every Banach space containing an isomorphic copy ofc0fails to have the fixed-point property for asymptotically nonexpansive mappings with respect to some locally convex topology which is coarser than the weak topology. If the copy ofc0is asymptotically isometric, this result can be improved, because we can prove the failure of the fixed-point property for nonexpansive mappings.

scholarly journals Usco selections of densely defined set-valued mappings

2002 ◽  
Vol 65 (2) ◽  
pp. 307-313 ◽  
Author(s):  
Warren B. Moors ◽  
Sivajah Somasundaram
Keyword(s):  
Baire Space ◽  
Topological Spaces ◽  
Range Space ◽  
Complete Space ◽  
Set Valued Mapping ◽  
Open Set ◽  
Set Valued Mappings ◽  
Densely Defined

A set-valued mapping Φ : X → 2Y acting between topological spaces X and Y is said to be “lower demicontinuous” if the interior of the closure of the set Φ−1(V): = {x ∈ X : Φ(x) ∩ V ≠ ∅} is dense in the closure of Φ−1(V) for each open set V in Y. Čoban, Kenderov and Revalski (1994) showed that for every densely defined lower demicontinuous mapping Φ acting from a Baire space X into subsets of a monotonely Čech-complete space Y, there exist a dense and Gδ subset X1 ⊆ X and an usco mapping G: X1 → 2Y such that G (x) ⊆ Φ*(x), for every x ∈ X1, where the mapping Φ*: X → 2Y is the extension of Φ defined by, W is a neighbourhood of x}.In this paper we present a proof of the above result with the notion of monotone Čcech-completeness replaced by the weaker notion of partition completeness. In addition, we observe that if the range space also lies is Stegall's class then we may assume that the mapping G is single-valued on X1.

scholarly journals Unconditionally converging polynomials on Banach spaces

1995 ◽  
Vol 117 (2) ◽  
pp. 321-331 ◽  
Author(s):  
Manuel Gonz´lez ◽  
Joaquín M. Gutiérrez
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Banach Spaces ◽  
Weak Topology ◽  
Linear Operators ◽  
Null Sequence ◽  
Good Tool ◽  
Image Position ◽  
Weakly Unconditionally Cauchy Series ◽  
Cauchy Series

In the study of polynomials acting on Banach spaces, the weak topology is not such a good tool as in the case of linear operators, due to the bad behaviour of the polynomials with respect to the weak convergence. For example,is a continuous polynomial taking a weakly null sequence into a sequence having no weakly Cauchy subsequences. In this paper we show that the situation is not so bad for unconditional series. Recall that is a weakly unconditionally Cauchy series (in short a w.u.C. series) in a Banach space E if for every f ε E* we have that and is an unconditionally converging series (in short an u.c. series) if every subseries is norm convergent.

Sign in / Sign up

Export Citation Format

Share Document