On sup- and inf-attaining functionals
Reflexivity is characterized by the weak-compactness of the unit ball. The weak-compactness of bounded, closed, and convex sets is characterized through sup-attaining functionals in view of the famous James’ theorem. The aim of this mathematical note is to provide the construction of bounded, closed...
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| Format: | Article |
| Language: | English |
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De Gruyter
2024-12-01
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| Series: | Open Mathematics |
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| Online Access: | https://doi.org/10.1515/math-2024-0090 |
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| _version_ | 1832570458696843264 |
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| author | García-Pacheco Francisco Javier |
| author_facet | García-Pacheco Francisco Javier |
| author_sort | García-Pacheco Francisco Javier |
| collection | DOAJ |
| description | Reflexivity is characterized by the weak-compactness of the unit ball. The weak-compactness of bounded, closed, and convex sets is characterized through sup-attaining functionals in view of the famous James’ theorem. The aim of this mathematical note is to provide the construction of bounded, closed, and convex subsets for which there exists a functional attaining its supremum on such a set but not its infimum. This construction leads to a characterization of reflexivity in the category of normed spaces and a characterization of full norm-attainment also in the category of normed spaces. |
| format | Article |
| id | doaj-art-c5472bb445af44b6a7f6ad1ab69f4813 |
| institution | Kabale University |
| issn | 2391-5455 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj-art-c5472bb445af44b6a7f6ad1ab69f48132025-02-02T15:46:01ZengDe GruyterOpen Mathematics2391-54552024-12-0122113313810.1515/math-2024-0090On sup- and inf-attaining functionalsGarcía-Pacheco Francisco Javier0Department of Mathematics, College of Engineering, University of Cadiz, Avenida de la Universidad 10, Puerto Real, SpainReflexivity is characterized by the weak-compactness of the unit ball. The weak-compactness of bounded, closed, and convex sets is characterized through sup-attaining functionals in view of the famous James’ theorem. The aim of this mathematical note is to provide the construction of bounded, closed, and convex subsets for which there exists a functional attaining its supremum on such a set but not its infimum. This construction leads to a characterization of reflexivity in the category of normed spaces and a characterization of full norm-attainment also in the category of normed spaces.https://doi.org/10.1515/math-2024-0090reflexivitynorm-attaining operatorbishop-phelps theoremsupporting vector46a0346a35 |
| spellingShingle | García-Pacheco Francisco Javier On sup- and inf-attaining functionals Open Mathematics reflexivity norm-attaining operator bishop-phelps theorem supporting vector 46a03 46a35 |
| title | On sup- and inf-attaining functionals |
| title_full | On sup- and inf-attaining functionals |
| title_fullStr | On sup- and inf-attaining functionals |
| title_full_unstemmed | On sup- and inf-attaining functionals |
| title_short | On sup- and inf-attaining functionals |
| title_sort | on sup and inf attaining functionals |
| topic | reflexivity norm-attaining operator bishop-phelps theorem supporting vector 46a03 46a35 |
| url | https://doi.org/10.1515/math-2024-0090 |
| work_keys_str_mv | AT garciapachecofranciscojavier onsupandinfattainingfunctionals |