{"title":"Banach空间直接和的球生成性质","authors":"Jan-David Hardtke","doi":"10.3318/PRIA.2015.115.13","DOIUrl":null,"url":null,"abstract":"A Banach space $X$ is said to have the ball generated property (BGP) if every closed, bounded, convex subset of $X$ can be written as an intersection of finite unions of closed balls. In 2002 S. Basu proved that the BGP is stable under (infinite) $c_0$- and $\\ell^p$-sums for $1<p<\\infty$. We will show here that for any absolute, normalised norm $\\|\\cdot\\|_E$ on $\\mathbb{R}^2$ satisfying a certain smoothness condition the direct sum $X\\oplus_E Y$ of two Banach spaces $X$ and $Y$ with respect to $\\|\\cdot\\|_E$ enjoys the BGP whenever $X$ and $Y$ have the BGP.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"52 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2015-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Ball generated property of direct sums of Banach spaces\",\"authors\":\"Jan-David Hardtke\",\"doi\":\"10.3318/PRIA.2015.115.13\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A Banach space $X$ is said to have the ball generated property (BGP) if every closed, bounded, convex subset of $X$ can be written as an intersection of finite unions of closed balls. In 2002 S. Basu proved that the BGP is stable under (infinite) $c_0$- and $\\\\ell^p$-sums for $1<p<\\\\infty$. We will show here that for any absolute, normalised norm $\\\\|\\\\cdot\\\\|_E$ on $\\\\mathbb{R}^2$ satisfying a certain smoothness condition the direct sum $X\\\\oplus_E Y$ of two Banach spaces $X$ and $Y$ with respect to $\\\\|\\\\cdot\\\\|_E$ enjoys the BGP whenever $X$ and $Y$ have the BGP.\",\"PeriodicalId\":434988,\"journal\":{\"name\":\"Mathematical Proceedings of the Royal Irish Academy\",\"volume\":\"52 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2015-02-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Proceedings of the Royal Irish Academy\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3318/PRIA.2015.115.13\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Proceedings of the Royal Irish Academy","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3318/PRIA.2015.115.13","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Ball generated property of direct sums of Banach spaces
A Banach space $X$ is said to have the ball generated property (BGP) if every closed, bounded, convex subset of $X$ can be written as an intersection of finite unions of closed balls. In 2002 S. Basu proved that the BGP is stable under (infinite) $c_0$- and $\ell^p$-sums for $1
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