Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces
Joint Authors
Source
Issue
Vol. 2019, Issue 2019 (31 Dec. 2019), pp.1-12, 12 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2019-05-02
Country of Publication
Egypt
No. of Pages
12
Main Subjects
Abstract EN
In this paper, we prove that if C⁎⁎ is a ε-separable bounded subset of X⁎⁎, then every convex function g≤σC is Ga^teaux differentiable at a dense Gδ subset G of X⁎ if and only if every subset of ∂σC(0)∩X is weakly dentable.
Moreover, we also prove that if C is a closed convex set, then dσC(x⁎)=x if and only if x is a weakly exposed point of C exposed by x⁎.
Finally, we prove that X is an Asplund space if and only if, for every bounded closed convex set C⁎ of X⁎, there exists a dense subset G of X⁎⁎ such that σC⁎ is Ga^teaux differentiable on G and dσC⁎(G)⊂C⁎.
We also prove that X is an Asplund space if and only if, for every w⁎-lower semicontinuous convex function f, there exists a dense subset G of X⁎⁎ such that f is Ga^teaux differentiable on G and df(G)⊂X⁎.
American Psychological Association (APA)
Shang, Shaoqiang& Cui, Yunan. 2019. Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces. Journal of Function Spaces،Vol. 2019, no. 2019, pp.1-12.
https://search.emarefa.net/detail/BIM-1174841
Modern Language Association (MLA)
Shang, Shaoqiang& Cui, Yunan. Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces. Journal of Function Spaces No. 2019 (2019), pp.1-12.
https://search.emarefa.net/detail/BIM-1174841
American Medical Association (AMA)
Shang, Shaoqiang& Cui, Yunan. Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces. Journal of Function Spaces. 2019. Vol. 2019, no. 2019, pp.1-12.
https://search.emarefa.net/detail/BIM-1174841
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-1174841