THE SET OF ELEMENTARY TENSORS IS WEAKLY CLOSED IN PROJECTIVE TENSOR PRODUCTS

IF 0.6 4区数学 Q3 MATHEMATICS Bulletin of the Australian Mathematical Society Pub Date : 2024-05-13 DOI:10.1017/s0004972724000376

COLIN PETITJEAN

{"title":"THE SET OF ELEMENTARY TENSORS IS WEAKLY CLOSED IN PROJECTIVE TENSOR PRODUCTS","authors":"COLIN PETITJEAN","doi":"10.1017/s0004972724000376","DOIUrl":null,"url":null,"abstract":"We prove that the set of elementary tensors is weakly closed in the projective tensor product of two Banach spaces. As a result, we answer a question of Rodríguez and Rueda Zoca [‘Weak precompactness in projective tensor products’, Indag. Math. (N.S.)35(1) (2024), 60–75], proving that if $(x_n) \\subset X$ and $(y_n) \\subset Y$ are two weakly null sequences such that $(x_n \\otimes y_n)$ converges weakly in $X \\widehat {\\otimes }_\\pi Y$ , then $(x_n \\otimes y_n)$ is also weakly null.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"1153 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the Australian Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s0004972724000376","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

We prove that the set of elementary tensors is weakly closed in the projective tensor product of two Banach spaces. As a result, we answer a question of Rodríguez and Rueda Zoca [‘Weak precompactness in projective tensor products’, Indag. Math. (N.S.)35(1) (2024), 60–75], proving that if

$(x_n) \subset X$

and

$(y_n) \subset Y$

are two weakly null sequences such that

$(x_n \otimes y_n)$

converges weakly in

$X \widehat {\otimes }_\pi Y$

, then

$(x_n \otimes y_n)$

is also weakly null.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

基本张量集合在射影张量积中是弱闭合的

我们证明了基本张量集合在两个巴拿赫空间的射影张量积中是弱封闭的。因此，我们回答了罗德里格斯（Rodríguez）和鲁埃达-佐卡（Rueda Zoca）的一个问题['射影张量积中的弱前封闭性'，Indag.(N.S.)35(1) (2024), 60-75)]，证明如果 $(x_n) \subset X$ 和 $(y_n) \subset Y$ 是两个弱空序列，使得 $(x_n \otimes y_n)$ 在 $X \widehat {\otimes }_\pi Y$ 中弱收敛，那么 $(x_n \otimes y_n)$ 也是弱空的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Bulletin of the Australian Mathematical Society 数学-数学

CiteScore

1.20

自引率

14.30%

发文量

149

审稿时长

4-8 weeks

期刊介绍： Bulletin of the Australian Mathematical Society aims at quick publication of original research in all branches of mathematics. Papers are accepted only after peer review but editorial decisions on acceptance or otherwise are taken quickly, normally within a month of receipt of the paper. The Bulletin concentrates on presenting new and interesting results in a clear and attractive way. Published Bi-monthly Published for the Australian Mathematical Society

期刊最新文献

EXTREMAL GRAPHS FOR DEGREE SUMS AND DOMINATING CYCLES GRAPHS WITH SEMITOTAL DOMINATION NUMBER HALF THEIR ORDER INEQUALITIES AND UNIFORM ASYMPTOTIC FORMULAE FOR SPT-CRANK OF PARTITIONS MONOGENIC EVEN QUARTIC TRINOMIALS ON A CONJECTURE ON SHIFTED PRIMES WITH LARGE PRIME FACTORS, II