A solution to the extreme point problem and other applications of Choquet theory to Lipschitz-free spaces
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911516433842176 |
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| author | Aliaga, Ramón J. Pernecká, Eva Smith, Richard J. |
| author_facet | Aliaga, Ramón J. Pernecká, Eva Smith, Richard J. |
| contents | We prove that every element of a Lipschitz-free space admits an expression as a convex series of elements with compact support. As a consequence, we conclude that all extreme points of the unit ball of Lipschitz-free spaces are elementary molecules, solving a long-standing problem. We also deduce that all elements of a Lipschitz-free space with the Radon-Nikodým property can be expressed as convex integrals of molecules. Our results are based on a recent theory of integral representation for functionals on Lipschitz spaces which draws on classical Choquet theory, due to the third named author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04312 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A solution to the extreme point problem and other applications of Choquet theory to Lipschitz-free spaces Aliaga, Ramón J. Pernecká, Eva Smith, Richard J. Functional Analysis 46B20, 46B04, 46E15 We prove that every element of a Lipschitz-free space admits an expression as a convex series of elements with compact support. As a consequence, we conclude that all extreme points of the unit ball of Lipschitz-free spaces are elementary molecules, solving a long-standing problem. We also deduce that all elements of a Lipschitz-free space with the Radon-Nikodým property can be expressed as convex integrals of molecules. Our results are based on a recent theory of integral representation for functionals on Lipschitz spaces which draws on classical Choquet theory, due to the third named author. |
| title | A solution to the extreme point problem and other applications of Choquet theory to Lipschitz-free spaces |
| topic | Functional Analysis 46B20, 46B04, 46E15 |
| url | https://arxiv.org/abs/2412.04312 |