巴拿赫空间的波兰空间:等距和同构类的复杂性

IF 1.1 2区 数学 Q1 MATHEMATICS
Marek Cúth, Martin Doležal, Michal Doucha, Ondřej Kurka
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For <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline2.png\" /> <jats:tex-math> $p\\in \\left [1,2\\right )\\cup \\left (2,\\infty \\right )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we show that the isometry classes of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline3.png\" /> <jats:tex-math> $L_p[0,1]$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline4.png\" /> <jats:tex-math> $\\ell _p$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline5.png\" /> <jats:tex-math> $G_\\delta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-complete sets and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline6.png\" /> <jats:tex-math> $F_{\\sigma \\delta }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-complete sets, respectively. Then we show that the isometry class of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline7.png\" /> <jats:tex-math> $c_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline8.png\" /> <jats:tex-math> $F_{\\sigma \\delta }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-complete set.</jats:p> <jats:p>Additionally, we compute the complexities of many other natural classes of separable Banach spaces; for instance, the class of separable <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline9.png\" /> <jats:tex-math> $\\mathcal {L}_{p,\\lambda +}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-spaces, for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline10.png\" /> <jats:tex-math> $p,\\lambda \\geq 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, is shown to be a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline11.png\" /> <jats:tex-math> $G_\\delta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-set, the class of superreflexive spaces is shown to be an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline12.png\" /> <jats:tex-math> $F_{\\sigma \\delta }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-set, and the class of spaces with local <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline13.png\" /> <jats:tex-math> $\\Pi $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-basis structure is shown to be a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline14.png\" /> <jats:tex-math> $\\boldsymbol {\\Sigma }^0_6$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-set. The paper is concluded with many open problems and suggestions for a future research.</jats:p>","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"54 1","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"POLISH SPACES OF BANACH SPACES: COMPLEXITY OF ISOMETRY AND ISOMORPHISM CLASSES\",\"authors\":\"Marek Cúth, Martin Doležal, Michal Doucha, Ondřej Kurka\",\"doi\":\"10.1017/s1474748023000440\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<jats:p>We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces, recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most classical separable Banach spaces.</jats:p> <jats:p>We prove that the infinite-dimensional separable Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinite-dimensional Banach space whose isomorphism class is <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1474748023000440_inline1.png\\\" /> <jats:tex-math> $F_\\\\sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. For <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1474748023000440_inline2.png\\\" /> <jats:tex-math> $p\\\\in \\\\left [1,2\\\\right )\\\\cup \\\\left (2,\\\\infty \\\\right )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we show that the isometry classes of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1474748023000440_inline3.png\\\" /> <jats:tex-math> $L_p[0,1]$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1474748023000440_inline4.png\\\" /> <jats:tex-math> $\\\\ell _p$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1474748023000440_inline5.png\\\" /> <jats:tex-math> $G_\\\\delta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-complete sets and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1474748023000440_inline6.png\\\" /> <jats:tex-math> $F_{\\\\sigma \\\\delta }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-complete sets, respectively. Then we show that the isometry class of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1474748023000440_inline7.png\\\" /> <jats:tex-math> $c_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1474748023000440_inline8.png\\\" /> <jats:tex-math> $F_{\\\\sigma \\\\delta }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-complete set.</jats:p> <jats:p>Additionally, we compute the complexities of many other natural classes of separable Banach spaces; for instance, the class of separable <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1474748023000440_inline9.png\\\" /> <jats:tex-math> $\\\\mathcal {L}_{p,\\\\lambda +}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-spaces, for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1474748023000440_inline10.png\\\" /> <jats:tex-math> $p,\\\\lambda \\\\geq 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, is shown to be a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1474748023000440_inline11.png\\\" /> <jats:tex-math> $G_\\\\delta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-set, the class of superreflexive spaces is shown to be an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1474748023000440_inline12.png\\\" /> <jats:tex-math> $F_{\\\\sigma \\\\delta }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-set, and the class of spaces with local <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1474748023000440_inline13.png\\\" /> <jats:tex-math> $\\\\Pi $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-basis structure is shown to be a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1474748023000440_inline14.png\\\" /> <jats:tex-math> $\\\\boldsymbol {\\\\Sigma }^0_6$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-set. The paper is concluded with many open problems and suggestions for a future research.</jats:p>\",\"PeriodicalId\":50002,\"journal\":{\"name\":\"Journal of the Institute of Mathematics of Jussieu\",\"volume\":\"54 1\",\"pages\":\"\"},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2023-11-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the Institute of Mathematics of Jussieu\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/s1474748023000440\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the Institute of Mathematics of Jussieu","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s1474748023000440","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

我们研究了巴拿赫空间的波兰空间中可分离巴拿赫空间的等距和同构类的复杂性,这些空间最近由作者在[14]中引入和研究。我们得到了关于最经典的可分离巴拿赫空间的尖锐结果。证明了无限维可分离Hilbert空间是唯一的等距类为闭的可分离无限维Banach空间,也是唯一的同构类为$F_\sigma $的可分离无限维Banach空间。对于$p\in \left [1,2\right )\cup \left (2,\infty \right )$,我们证明了$L_p[0,1]$和$\ell _p$的等距类分别是$G_\delta $ -完备集和$F_{\sigma \delta }$ -完备集。然后证明了$c_0$的等距类是一个$F_{\sigma \delta }$完备集。此外,我们计算了许多其他自然类别的可分离Banach空间的复杂性;例如,对于$p,\lambda \geq 1$,可分离的$\mathcal {L}_{p,\lambda +}$ -空间类被证明是一个$G_\delta $ -集,超自反的空间类被证明是一个$F_{\sigma \delta }$ -集,具有局部$\Pi $ -基结构的空间类被证明是一个$\boldsymbol {\Sigma }^0_6$ -集。文章最后提出了许多有待解决的问题和对未来研究的建议。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
POLISH SPACES OF BANACH SPACES: COMPLEXITY OF ISOMETRY AND ISOMORPHISM CLASSES
We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces, recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most classical separable Banach spaces. We prove that the infinite-dimensional separable Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinite-dimensional Banach space whose isomorphism class is $F_\sigma $ . For $p\in \left [1,2\right )\cup \left (2,\infty \right )$ , we show that the isometry classes of $L_p[0,1]$ and $\ell _p$ are $G_\delta $ -complete sets and $F_{\sigma \delta }$ -complete sets, respectively. Then we show that the isometry class of $c_0$ is an $F_{\sigma \delta }$ -complete set. Additionally, we compute the complexities of many other natural classes of separable Banach spaces; for instance, the class of separable $\mathcal {L}_{p,\lambda +}$ -spaces, for $p,\lambda \geq 1$ , is shown to be a $G_\delta $ -set, the class of superreflexive spaces is shown to be an $F_{\sigma \delta }$ -set, and the class of spaces with local $\Pi $ -basis structure is shown to be a $\boldsymbol {\Sigma }^0_6$ -set. The paper is concluded with many open problems and suggestions for a future research.
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来源期刊
CiteScore
2.40
自引率
0.00%
发文量
54
审稿时长
>12 weeks
期刊介绍: The Journal of the Institute of Mathematics of Jussieu publishes original research papers in any branch of pure mathematics; papers in logic and applied mathematics will also be considered, particularly when they have direct connections with pure mathematics. Its policy is to feature a wide variety of research areas and it welcomes the submission of papers from all parts of the world. Selection for publication is on the basis of reports from specialist referees commissioned by the Editors.
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