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.

Existence Of Nearest Points In Banach Spaces

1989 ◽  
Vol 41 (4) ◽  
pp. 702-720 ◽  
Author(s):  
Jonathan M. Borwein ◽  
Simon Fitzpatrick
Keyword(s):  
Banach Spaces ◽  
Closed Subset ◽  
Extension Theorem ◽  
Closed Set ◽  
Reflexive Space ◽  
Nearest Point ◽  
Open Question ◽  
Nearest Points

This paper makes a unified development of what the authors know about the existence of nearest points to closed subsets of (real) Banach spaces. Our work is made simpler by the methodical use of subderivatives. The results of Section 3 and Section 7 in particular are, to the best of our knowledge, new. In Section 5 and Section 6 we provide refined proofs of the Lau-Konjagin nearest point characterizations of reflexive Kadec spaces (Theorem 5.11, Theorem 6.6) and give a substantial extension (Theorem 5.12). The main open question is: are nearest points dense in the boundary of every closed subset of every reflexive space? Indeed can a proper closed set in a reflexive space fail to have any nearest points? In Section 7 we show that there are some non-Kadec reflexive spaces in which nearest points are dense in the boundary of every closed set.

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 to closed sets and directional derivatives of distance functions

1989 ◽  
Vol 39 (2) ◽  
pp. 233-238 ◽  
Author(s):  
Simon Fitzpatrick
Keyword(s):  
Distance Function ◽  
Directional Derivative ◽  
Distance Functions ◽  
Directional Derivatives ◽  
Closed Set ◽  
Closed Sets ◽  
Set Of Points ◽  
Derivatives Of ◽  
Nearest Points ◽  
Dense Set

We investigate the circumstances under which the distance function to a closed set in a Banach space having a one-sided directional derivative equal to 1 or −1 implies the existence of nearest points. In reflexive spaces we show that at a dense set of points outside a closed set the distance function has a directional derivative equal to 1.

scholarly journals Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces

1993 ◽  
Vol 47 (2) ◽  
pp. 205-212 ◽  
Author(s):  
J.R. Giles ◽  
Scott Sciffer
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Lipschitz Function ◽  
Separable Banach Space ◽  
Locally Lipschitz Function ◽  
Locally Lipschitz Functions ◽  
Dini Derivative ◽  
Locally Lipschitz ◽  
Set Of Points ◽  
Clarke Derivative

For a locally Lipschitz function on a separable Banach space the set of points of Gâteaux differentiability is dense but not necessarily residual. However, the set of points where the upper Dini derivative and the Clarke derivative agree is residual. It follows immediately that the set of points of intermediate differentiability is also residual and the set of points where the function is Gâteaux but not strictly differentiable is of the first category.

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 D-gap Functions for a Class of Equilibrium Problems in Banach Spaces

2003 ◽  
Vol 3 (2) ◽  
pp. 274-286 ◽  
Author(s):  
I. V. Konnov ◽  
O. V. Pinyagina
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Stationary Point ◽  
Convex Functions ◽  
General Class ◽  
Real Banach Space ◽  
Equilibrium Problems ◽  
Gap Functions ◽  
Constrained Equilibrium ◽  
Type Algorithm

AbstractWe consider rather a general class of equilibrium problems in a real Banach space, which involve nonsmooth convex functions. We apply the D-gap function approach to these problems and show that, under certain additional assumptions, they can be converted into a problem of finding a stationary point of a differentiable function. Based on this property, we suggest a descent type algorithm to find a solution to the initial problem. An example of applications to nonlinearly constrained equilibrium problems is also given.

Strict U-Ideals and U-Summands in Banach Spaces

2014 ◽  
Vol 114 (2) ◽  
pp. 216 ◽  
Author(s):  
Trond A. Abrahamsen
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Extension Operator ◽  
Denting Points ◽  
Set Of Points

For a strict u-ideal $X$ in a Banach space $Y$ we show that the set of points in the dual unit ball $B_{X^{\ast}}$, strongly exposed by points in the range $\it TY$ of the unconditional extension operator $T$ from $Y$ into the bidual $X^{\ast\ast}$ of $X$, is contained in the weak$^{\ast}$ denting points in $B_{X^{\ast}}$. We also prove that a u-embedded space is a u-summand if and only if it contains no copy of $c_0$ if and only if it is weakly sequentially complete.

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 Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 150
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Dimensional Algebra ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Linear Subspaces ◽  
Infinite Dimensional Banach Space

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

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