A Minimax Theorem for L-0-Valued Functions on Random Normed Modules
We generalize the well-known minimax theorems to L¯0-valued functions on random normed modules. We first give some basic properties of an L0-valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, the (ε,λ)-topology and the locally L0-convex topology...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2013/704251 |
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| author | Shien Zhao Yuan Zhao |
| author_facet | Shien Zhao Yuan Zhao |
| author_sort | Shien Zhao |
| collection | DOAJ |
| description | We generalize the well-known minimax theorems to L¯0-valued functions on random normed modules. We first give some basic properties of an L0-valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, the (ε,λ)-topology and the locally L0-convex topology. Then, we introduce the definition of random saddle points. Conditions for an L0-valued function to have a random saddle point are given. The most greatest difference between our results and the classical minimax theorems is that we have to overcome the difficulty resulted from the lack of the condition of compactness. Finally, we, using relations between the two kinds of topologies, establish the minimax theorem of L¯0-valued functions in the framework of random normed modules and random conjugate spaces. |
| format | Article |
| id | doaj-art-dcd06e5535da4825a2ddcee39a4eeabe |
| institution | OA Journals |
| issn | 0972-6802 1758-4965 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces and Applications |
| spelling | doaj-art-dcd06e5535da4825a2ddcee39a4eeabe2025-08-20T02:21:38ZengWileyJournal of Function Spaces and Applications0972-68021758-49652013-01-01201310.1155/2013/704251704251A Minimax Theorem for L-0-Valued Functions on Random Normed ModulesShien Zhao0Yuan Zhao1Elementary Educational College, Capital Normal University, Beijing 100048, ChinaDepartment of Basic Sciences, Hebei Finance University, Baoding 071051, ChinaWe generalize the well-known minimax theorems to L¯0-valued functions on random normed modules. We first give some basic properties of an L0-valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, the (ε,λ)-topology and the locally L0-convex topology. Then, we introduce the definition of random saddle points. Conditions for an L0-valued function to have a random saddle point are given. The most greatest difference between our results and the classical minimax theorems is that we have to overcome the difficulty resulted from the lack of the condition of compactness. Finally, we, using relations between the two kinds of topologies, establish the minimax theorem of L¯0-valued functions in the framework of random normed modules and random conjugate spaces.http://dx.doi.org/10.1155/2013/704251 |
| spellingShingle | Shien Zhao Yuan Zhao A Minimax Theorem for L-0-Valued Functions on Random Normed Modules Journal of Function Spaces and Applications |
| title | A Minimax Theorem for L-0-Valued Functions on Random Normed Modules |
| title_full | A Minimax Theorem for L-0-Valued Functions on Random Normed Modules |
| title_fullStr | A Minimax Theorem for L-0-Valued Functions on Random Normed Modules |
| title_full_unstemmed | A Minimax Theorem for L-0-Valued Functions on Random Normed Modules |
| title_short | A Minimax Theorem for L-0-Valued Functions on Random Normed Modules |
| title_sort | minimax theorem for l 0 valued functions on random normed modules |
| url | http://dx.doi.org/10.1155/2013/704251 |
| work_keys_str_mv | AT shienzhao aminimaxtheoremforl0valuedfunctionsonrandomnormedmodules AT yuanzhao aminimaxtheoremforl0valuedfunctionsonrandomnormedmodules AT shienzhao minimaxtheoremforl0valuedfunctionsonrandomnormedmodules AT yuanzhao minimaxtheoremforl0valuedfunctionsonrandomnormedmodules |