原子映射空间中的广义Hahn-Banach定理

Generalized Hahn-Banach Theorem in Nuclear Mapping Spaces

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.