A Banach space whose set of norm-attaining functionals is algebraically trivial
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910774807494656 |
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| author | Martin, Miguel |
| author_facet | Martin, Miguel |
| contents | We construct a Banach space $X$ for which the set of norm-attaining functionals $NA(X,\mathbb{R})$ does not contain any non-trivial cone. Even more, given two linearly independent norm-attaining functionals on $X$, no other element of the segment between them attains its norm. Equivalently, the intersection of $NA(X,\mathbb{R})$ with a two-dimensional subspace of $X^*$ is contained in the union of two lines. In terms of proximinality, we show that for every closed subspace $M$ of $X$ of codimension two, at most four elements of the unit sphere of $X/M$ have a representative of norm-one. We further relate this example with an open problem on norm-attaining operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07273 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Banach space whose set of norm-attaining functionals is algebraically trivial Martin, Miguel Functional Analysis Primary 46B04, 46B20, Secondary 41A65, 46B03, 46B22, 46B87 We construct a Banach space $X$ for which the set of norm-attaining functionals $NA(X,\mathbb{R})$ does not contain any non-trivial cone. Even more, given two linearly independent norm-attaining functionals on $X$, no other element of the segment between them attains its norm. Equivalently, the intersection of $NA(X,\mathbb{R})$ with a two-dimensional subspace of $X^*$ is contained in the union of two lines. In terms of proximinality, we show that for every closed subspace $M$ of $X$ of codimension two, at most four elements of the unit sphere of $X/M$ have a representative of norm-one. We further relate this example with an open problem on norm-attaining operators. |
| title | A Banach space whose set of norm-attaining functionals is algebraically trivial |
| topic | Functional Analysis Primary 46B04, 46B20, Secondary 41A65, 46B03, 46B22, 46B87 |
| url | https://arxiv.org/abs/2406.07273 |