Computability of the Hahn-Banach Theorem Revisited
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917350308052992 |
|---|---|
| author | Brattka, Vasco Sorg, Christopher |
| author_facet | Brattka, Vasco Sorg, Christopher |
| contents | Computational properties of the Hahn-Banach theorem have been studied in computable, constructive and reverse mathematics and in all these approaches the theorem is equivalent to weak Kőnig's lemma. Gherardi and Marcone proved that this is also true in the uniform sense of Weihrauch complexity. However, their result requires the underlying space to be variable. We prove that the Hahn-Banach theorem attains its full complexity already for the Banach space $\ell^1$. We also prove that the one-step Hahn-Banach theorem for this space is Weihrauch equivalent to the intermediate value theorem. This also yields a new and very simple proof of the reduction of the Hahn-Banach theorem to weak Kőnig's lemma using infinite products. Finally, we show that the Hahn-Banach theorem for $\ell^1$ in the two-dimensional case is Weihrauch equivalent to the lesser limited principle of omniscience. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16802 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Computability of the Hahn-Banach Theorem Revisited Brattka, Vasco Sorg, Christopher Logic Logic in Computer Science 03F60 (Primary) 03D78, 46S30 (Secondary) F.4.1; F.2.1 Computational properties of the Hahn-Banach theorem have been studied in computable, constructive and reverse mathematics and in all these approaches the theorem is equivalent to weak Kőnig's lemma. Gherardi and Marcone proved that this is also true in the uniform sense of Weihrauch complexity. However, their result requires the underlying space to be variable. We prove that the Hahn-Banach theorem attains its full complexity already for the Banach space $\ell^1$. We also prove that the one-step Hahn-Banach theorem for this space is Weihrauch equivalent to the intermediate value theorem. This also yields a new and very simple proof of the reduction of the Hahn-Banach theorem to weak Kőnig's lemma using infinite products. Finally, we show that the Hahn-Banach theorem for $\ell^1$ in the two-dimensional case is Weihrauch equivalent to the lesser limited principle of omniscience. |
| title | Computability of the Hahn-Banach Theorem Revisited |
| topic | Logic Logic in Computer Science 03F60 (Primary) 03D78, 46S30 (Secondary) F.4.1; F.2.1 |
| url | https://arxiv.org/abs/2603.16802 |