Schauder Bases Having Many Good Block Basic Sequences
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2019
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| _version_ | 1866915092732313600 |
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| author | Krause, Cory A. |
| author_facet | Krause, Cory A. |
| contents | In the study of asymptotic geometry in Banach spaces, a basic sequence which gives rise to a spreading model has been called a good sequence. It is well known that every normalized basic sequence in a Banach space has a subsequence which is good. We investigate the assumption that every normalized block tree relative to a basis has a branch which is good. This combinatorial property turns out to be very strong and is equivalent to the space being $1$-asymptotic $\ell_p$ for some $1\leq p\leq\infty$. We also investigate the even stronger assumption that every block basic sequence of a basis is good. Finally, using the Hindman-Milliken-Taylor theorem, we prove a stabilization theorem which produces a basic sequence all of whose normalized constant coefficient block basic sequences are good, and we present an application of this stabilization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1907_11863 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Schauder Bases Having Many Good Block Basic Sequences Krause, Cory A. Functional Analysis Primary 46B03, 46B06, 46B25, 46B45, Secondary 05D10 In the study of asymptotic geometry in Banach spaces, a basic sequence which gives rise to a spreading model has been called a good sequence. It is well known that every normalized basic sequence in a Banach space has a subsequence which is good. We investigate the assumption that every normalized block tree relative to a basis has a branch which is good. This combinatorial property turns out to be very strong and is equivalent to the space being $1$-asymptotic $\ell_p$ for some $1\leq p\leq\infty$. We also investigate the even stronger assumption that every block basic sequence of a basis is good. Finally, using the Hindman-Milliken-Taylor theorem, we prove a stabilization theorem which produces a basic sequence all of whose normalized constant coefficient block basic sequences are good, and we present an application of this stabilization. |
| title | Schauder Bases Having Many Good Block Basic Sequences |
| topic | Functional Analysis Primary 46B03, 46B06, 46B25, 46B45, Secondary 05D10 |
| url | https://arxiv.org/abs/1907.11863 |