Almost Auerbach, Markushevich and Schauder bases in Hilbert and Banach spaces
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911930553204736 |
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| author | Tselishchev, Anton |
| author_facet | Tselishchev, Anton |
| contents | For any sequence of positive numbers $(\varepsilon_n)_{n=1}^\infty$ such that $\sum_{n=1}^\infty \varepsilon_n = \infty$ we provide an explicit simple construction of $(1+\varepsilon_n)$-bounded Markushevich basis in a separable Hilbert space which is not strong, or, in other terminology, is not hereditary complete; this condition on the sequence $(\varepsilon_n)_{n=1}^\infty$ is sharp. Using a finite-dimensional version of this construction, Dvoretzky's theorem and a construction of Vershynin, we conclude that in any Banach space for any sequence of positive numbers $(\varepsilon_n)_{n=1}^\infty$ such that $\sum_{n=1}^\infty \varepsilon_n^2 = \infty$ there exists a $(1+\varepsilon_n)$-bounded Markushevich basis which is not a Schauder basis after any permutation of its elements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16467 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Almost Auerbach, Markushevich and Schauder bases in Hilbert and Banach spaces Tselishchev, Anton Functional Analysis 46B15, 46B04, 46C15 For any sequence of positive numbers $(\varepsilon_n)_{n=1}^\infty$ such that $\sum_{n=1}^\infty \varepsilon_n = \infty$ we provide an explicit simple construction of $(1+\varepsilon_n)$-bounded Markushevich basis in a separable Hilbert space which is not strong, or, in other terminology, is not hereditary complete; this condition on the sequence $(\varepsilon_n)_{n=1}^\infty$ is sharp. Using a finite-dimensional version of this construction, Dvoretzky's theorem and a construction of Vershynin, we conclude that in any Banach space for any sequence of positive numbers $(\varepsilon_n)_{n=1}^\infty$ such that $\sum_{n=1}^\infty \varepsilon_n^2 = \infty$ there exists a $(1+\varepsilon_n)$-bounded Markushevich basis which is not a Schauder basis after any permutation of its elements. |
| title | Almost Auerbach, Markushevich and Schauder bases in Hilbert and Banach spaces |
| topic | Functional Analysis 46B15, 46B04, 46C15 |
| url | https://arxiv.org/abs/2406.16467 |