scholarly journals Behavior of holomorphic mappings on $p$-compact sets in a Banach space

2015 ◽  
Vol 368 (7) ◽  
pp. 4855-4871 ◽  
Author(s):  
Richard M. Aron ◽  
Erhan Çalışkan ◽  
Domingo García ◽  
Manuel Maestre
Keyword(s):  
Banach Space ◽  
Holomorphic Mappings ◽  
Compact Sets

Banach Spaces that are Uniformly Rotund in Weakly Compact Sets of Directions

1977 ◽  
Vol 29 (5) ◽  
pp. 963-970 ◽  
Author(s):  
Mark A. Smith
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Nonzero Element ◽  
Bounded Subset ◽  
Compact Sets ◽  
Weakly Compact ◽  
Uniform Rotundity ◽  
Minimum Depth ◽  
Weakly Compact Sets

In a Banach space, the directional modulus of rotundity, δ (ϵ, z), measures the minimum depth at which the midpoints of all chords of the unit ball which are parallel to z and of length at least ϵ are buried beneath the surface. A Banach space is uniformly rotund in every direction (URED) if δ (ϵ, z) is positive for every positive ϵ and every nonzero element z. This concept of directionalized uniform rotundity was introduced by Garkavi [6] to characterize those Banach spaces in which every bounded subset has at most one Čebyšev center.

scholarly journals Schwarz-Pick Estimates for Holomorphic Mappings with Values in Homogeneous Ball

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Jianfei Wang
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Arbitrary Order ◽  
Complex Banach Space ◽  
Holomorphic Mappings ◽  
Partial Derivatives ◽  
Carathéodory Metric ◽  
Homogeneous Ball ◽  
Derivatives Of

LetBXbe the unit ball in a complex Banach spaceX. AssumeBXis homogeneous. The generalization of the Schwarz-Pick estimates of partial derivatives of arbitrary order is established for holomorphic mappings from the unit ballBntoBXassociated with the Carathéodory metric, which extend the corresponding Chen and Liu, Dai et al. results.

scholarly journals Mazur's intersection property of balls for compact convex sets

1987 ◽  
Vol 35 (2) ◽  
pp. 267-274 ◽  
Author(s):  
J. H. M. Whitfield ◽  
V. Zizler
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Intersection Property ◽  
Compact Sets ◽  
Compact Convex Set ◽  
Mazur's Intersection Property

We show that every compact convex set in a Banach space X is an intersection of balls provided the cone generated by the set of all extreme points of the dual unit ball of X* is dense in X* in the topology of uniform convergence on compact sets in X. This allows us to renorm every Banach space with transfinite Schauder basis by a norm which shares the mentioned intersection property.

scholarly journals Farthest points and the farthest distance map

2005 ◽  
Vol 71 (3) ◽  
pp. 425-433 ◽  
Author(s):  
Pradipta Bandyopadhyay ◽  
S. Dutta
Keyword(s):  
Banach Space ◽  
Maximal Monotone ◽  
Convex Banach Space ◽  
Closed Convex Hull ◽  
Compact Sets ◽  
Weakly Compact ◽  
Distance Map ◽  
Farthest Points ◽  
Bounded Set ◽  
Weakly Compact Sets

In this paper, we consider farthest points and the farthest distance map of a closed bounded set in a Banach space. We show, inter alia, that a strictly convex Banach space has the Mazur intersection property for weakly compact sets if and only if every such set is the closed convex hull of its farthest points, and recapture a classical result of Lau in a broader set-up. We obtain an expression for the subdifferential of the farthest distance map in the spirit of Preiss' Theorem which in turn extends a result of Westphal and Schwartz, showing that the subdifferential of the farthest distance map is the unique maximal monotone extension of a densely defined monotone operator involving the duality map and the farthest point map.

HARTOGS-TYPE EXTENSION THEOREM AND SINGULAR SETS OF SEPARATELY HOLOMORPHIC MAPPINGS ON COMPACT SETS WITH VALUES IN A WEAKLY BRODY HYPERBOLIC COMPLEX SPACE

2001 ◽  
Vol 12 (07) ◽  
pp. 857-865
Author(s):  
DO DUC THAI ◽  
NGUYEN THI TUYET MAI
Keyword(s):  
Complex Space ◽  
Extension Theorem ◽  
Holomorphic Mappings ◽  
Compact Sets ◽  
Singular Sets ◽  
Hyperbolic Complex

We give a Hartogs-type extension theorem for separately holomorphic mappings on compact sets into a weakly Brody hyperbolic complex space. Moreover, a generalization of Saint Raymond–Siciak theorem of the singular sets of separately holomorphic mappings with values in a weakly Brody hyperbolic complex space is given.

THE BOUNDED APPROXIMATION PROPERTY FOR THE WEIGHTED SPACES OF HOLOMORPHIC MAPPINGS ON BANACH SPACES

2017 ◽  
Vol 60 (2) ◽  
pp. 307-320 ◽  
Author(s):  
MANJUL GUPTA ◽  
DEEPIKA BAWEJA
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Open Subset ◽  
Weighted Space ◽  
Approximation Property ◽  
Holomorphic Mappings ◽  
Countable Family ◽  
Weighted Spaces ◽  
Metric Approximation

AbstractIn this paper, we study the bounded approximation property for the weighted space$\mathcal{HV}$(U) of holomorphic mappings defined on a balanced open subsetUof a Banach spaceEand its predual$\mathcal{GV}$(U), where$\mathcal{V}$is a countable family of weights. After obtaining an$\mathcal{S}$-absolute decomposition for the space$\mathcal{GV}$(U), we show thatEhas the bounded approximation property if and only if$\mathcal{GV}$(U) has. In case$\mathcal{V}$consists of a single weightv, an analogous characterization for the metric approximation property for a Banach spaceEhas been obtained in terms of the metric approximation property for the space$\mathcal{G}_v$(U).

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