{"title":"Banach空间的渐近光滑性与普适性","authors":"R. Causey, G. Lancien","doi":"10.4064/cm8923-12-2022","DOIUrl":null,"url":null,"abstract":"For $1<p\\leqslant \\infty$, we study the complexity and the existence of universal spaces for two classes of separable Banach spaces, denoted $\\textsf{A}_p$ and $\\textsf{N}_p$, and related to asymptotic smoothness in Banach spaces. We show that each of these classes is Borel in the class of separable Banach spaces. Then we build small families of Banach spaces that are both injectively and surjectively universal for these classes. Finally, we prove the optimality of this universality result, by proving in particular that none of these classes admits a universal space.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2022-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Asymptotic smoothness and universality\\nin Banach spaces\",\"authors\":\"R. Causey, G. Lancien\",\"doi\":\"10.4064/cm8923-12-2022\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For $1<p\\\\leqslant \\\\infty$, we study the complexity and the existence of universal spaces for two classes of separable Banach spaces, denoted $\\\\textsf{A}_p$ and $\\\\textsf{N}_p$, and related to asymptotic smoothness in Banach spaces. We show that each of these classes is Borel in the class of separable Banach spaces. Then we build small families of Banach spaces that are both injectively and surjectively universal for these classes. Finally, we prove the optimality of this universality result, by proving in particular that none of these classes admits a universal space.\",\"PeriodicalId\":49216,\"journal\":{\"name\":\"Colloquium Mathematicum\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2022-07-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Colloquium Mathematicum\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4064/cm8923-12-2022\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Colloquium Mathematicum","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4064/cm8923-12-2022","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
期刊介绍:
Colloquium Mathematicum is a journal devoted to the publication of original papers of moderate length addressed to a broad mathematical audience. It publishes results of original research, interesting new proofs of important theorems and research-expository papers in all fields of pure mathematics.
Two issues constitute a volume, and at least four volumes are published each year.