Differentiability of convex functions and the convex point of continuity property in Banach spaces

1987 ◽  
Vol 59 (2) ◽  
pp. 245-255 ◽  
Author(s):  
R. Deville ◽  
G. Godefroy ◽  
D. E. G. Hare ◽  
V. Zizler
Keyword(s):  
Banach Spaces ◽  
Convex Functions ◽  
Continuity Property ◽  
Point Of Continuity Property ◽  
Convex Point

A Dual characterization of Banach Spaces With the Convex Point-of-Continuity Property

1989 ◽  
Vol 32 (3) ◽  
pp. 274-280
Author(s):  
D. E. G. Hare
Keyword(s):  
Banach Spaces ◽  
Continuity Property ◽  
Dual Spaces ◽  
Equivalent Norms ◽  
Fréchet Differentiability ◽  
New Type ◽  
Frechet Differentiability ◽  
Point Of Continuity Property ◽  
Convex Point

AbstractWe introduce a new type of differentiability, called cofinite Fréchet differentiability. We show that the convex point-of-continuity property of Banach spaces is dual to the cofinite Fréchet differentiability of all equivalent norms. A corresponding result for dual spaces with the weak* convex point-of-continuity property is also established.

scholarly journals Phelps spaces and finite dimensional decompositions

1988 ◽  
Vol 37 (2) ◽  
pp. 263-271 ◽  
Author(s):  
R. Deville ◽  
G. Godefroy ◽  
D.E.G. Hare ◽  
V. Zizler
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Continuity Property ◽  
Finite Dimensional ◽  
Point Of Continuity Property ◽  
Convex Point

We show that if X is a separable Banach space such that X* fails the weak* convex point-of-continuity property (C*PCP), then there is a subspace Y of X such that both Y* and (X/Y)* fail C*PCP and both Y and X/Y have finite dimensional Schauder decompositions.

The point of continuity property in Banach spaces not containing ℓ1

2011 ◽  
Vol 191 (1) ◽  
pp. 347-361 ◽  
Author(s):  
Ginés López Pérez ◽  
José A. Soler Arias
Keyword(s):  
Banach Spaces ◽  
Continuity Property ◽  
Point Of Continuity Property

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.

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