原子映射空间中的广义Hahn-Banach定理  

Generalized Hahn-Banach Theorem in Nuclear Mapping Spaces

作　　者：董平川 董浙[2] 姜海益[2] DONG Pingchuan;DONG Zhe;JIANG Haiyi(Department of Mathematics,New York University,New York,NY 10012-1110;School of Mathematical Sciences,Zhejiang University,Hangzhou 310027,China)

机构地区：[1]纽约大学数学系,纽约10012-1110 [2]浙江大学数学科学学院,杭州310027

出　　处：《数学年刊（A辑）》2020年第4期399-408,共10页Chinese Annals of Mathematics

基　　金：国家自然科学基金(No.11871423)的资助。

摘　　要：经典的Hahn-Banach定理告诉读者在有界映射空间(B(·,·),‖·‖)中C具有内射性.在第二节中主要研究在原子映射空间(N^(B)(·,·),v^(B))中的内射性.作者得到任意有限维Banach空间在原子映射空间(N^(B)(·,),v^(B))中都是内射的.这可以看作(N^(B)(·,·),v^(B))中的广义Hahn-Banach定理.在经典的Banach空间理论中,众所周知一个Banach空间E在(B(·,·),‖·‖)中具有{l1n}_(n∈N)有限可表示性当且仅当E同构于某个超积∏l1n(α)的子空间.作为第二节的一个应用,第三节中作者研究了在原子映射空间(N^(B)(·,·),v^(B))中的{l1n}_(n∈N)有限可表示性.作者得到C是唯一在原子映射空间(N^(B)(·,·),v^(B))中具有{l1n}_(n∈N)有限可表示性的Banach空间.这与Banach空间理论中的经典结果是迥然不同的.Classical Hahn-Banach theorem says that C is injective in the system of bounded mapping spaces(B(·,·),‖·‖).It is the key initial ingredient of functional analysis.In Section2 the authors mainly investigate its analogue in the system of nuclear mapping spaces(N^(B)(·,·),v^(B)).The authors obtain that any finite-dimensional Banach space is injective in the system(N^(B)(·,·),v^(B)).This can be considered as the generalized Hahn-Banach theorem in the system(N^(B)(·,·),v^(B)).In the classical Banach space theory,a Banach space E is finitely representable in{l1n}_(n∈N)in the system(B(·,·),‖·‖)if and only if E is isometric to a subspace of some ultraproduct∏l1n(α).As one interesting application of Section 2,in Section 3 they study the finite representability in{l1n}_(n∈N)in the system(N^(B)(·,·),v^(B)).They obtain that C is the unique Banach space which is finitely representable in{l1n}_(n∈N)in the system(N^(B)(·,·),v^(B)).This is quite strange and different from the classical result in Banach space theory.

关 键 词：HAHN-BANACH定理 原子映射空间 内射性 有限可表示性 

分 类 号：O175.29[理学—数学]