Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces
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...
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| Format: | Article |
| Language: | English |
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Wiley
2019-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2019/6852859 |
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| author | Shaoqiang Shang Yunan Cui |
| author_facet | Shaoqiang Shang Yunan Cui |
| author_sort | Shaoqiang Shang |
| collection | DOAJ |
| description | 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⁎. |
| format | Article |
| id | doaj-art-a0ca0f4559184a84a57e14698209d9ca |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2019-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-a0ca0f4559184a84a57e14698209d9ca2025-02-03T01:32:46ZengWileyJournal of Function Spaces2314-88962314-88882019-01-01201910.1155/2019/68528596852859Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach SpacesShaoqiang Shang0Yunan Cui1Academy of Mathematical Sciences, Harbin Engineering University, Harbin 150001, ChinaDepartment of Mathematics, Harbin University of Science and Technology University, Harbin 150080, ChinaIn 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⁎.http://dx.doi.org/10.1155/2019/6852859 |
| spellingShingle | Shaoqiang Shang Yunan Cui Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces Journal of Function Spaces |
| title | Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces |
| title_full | Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces |
| title_fullStr | Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces |
| title_full_unstemmed | Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces |
| title_short | Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces |
| title_sort | gateaux differentiability of convex functions and weak dentable set in nonseparable banach spaces |
| url | http://dx.doi.org/10.1155/2019/6852859 |
| work_keys_str_mv | AT shaoqiangshang gateauxdifferentiabilityofconvexfunctionsandweakdentablesetinnonseparablebanachspaces AT yunancui gateauxdifferentiabilityofconvexfunctionsandweakdentablesetinnonseparablebanachspaces |