董平川,董 浙,姜海益.原子映射空间中的广义Hahn-Banach定理[J].数学年刊A辑,2020,(4):399~408
原子映射空间中的广义Hahn-Banach定理
Generalized Hahn-Banach Theorem in Nuclear Mapping Spaces
Received:February 27, 2019  Revised:March 29, 2020
DOI:10.16205/j.cnki.cama.2020.0028
中文关键词:  Hahn-Banach定理, 原子映射空间, 内射性, 有限可表示性
英文关键词:Hahn-Banach theorem, Nuclear mapping space, Injectivity, Finite representability
基金项目:国家自然科学基金(No.11871423)
Author NameAffiliation
DONG Pingchuan Department of Mathematics, New York University, New York, NY 10012-1110,USA. 
DONG Zhe Corresponding Author. School of Mathematical Sciences, Zhejiang University,Hangzhou 310027, China. 
JIANG Haiyi School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China. 
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中文摘要:
      经典的Hahn-Banach定理告诉读者在有界映射空间(B(.,.), \|\cdot\|)中\mathbb{C具有内射性. 在第二节中主要研究在原子映射空间(\n^{B}(\cdot, \cdot), \nu^{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}(\cdot, \cdot), \nu^{B})中都是内射的. 这可以看作(\n^{B}(\cdot, \cdot), \nu^{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(\cdot, \cdot), \|\cdot\|)中具有\{\ell_{1}^{n}\}_{n\in\mathbb{N}有限可表示性当且仅当E同构于某个超积\prod\ell_{1}^{n(\alpha)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}(\cdot, \cdot), \nu^{B})中的\{\ell_{1}^{n}\}_{n\in\mathbb{N}有限可表示性. 作者得到 \mathbb{C是唯一在原子映射空间(\n^{B}(\cdot, \cdot), \nu^{B})中具有\{\ell_{1}^{n}\}_{n\in\mathbb{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.
英文摘要:
      lassical Hahn-Banach theorem says that C is injective in the system of bounded mapping spaces (B(·, ·), k·k). It is the key initial ingredient of functional analysis. In Section 2 the authors mainly investigate its analogue in the system of nuclear mapping spaces 408 t + D A v 41  (N B(·, ·), νB). The authors obtain that any finite-dimensional Banach space is injective in the system (N B(·, ·), νB). This can be considered as the generalized Hahn-Banach theorem in the system (N B(·, ·), νB).
In the classical Banach space theory, a Banach space E is finitely representable in {?n1 }n∈N in the system (B(·, ·), k · k) if and only if E is isometric to a subspace of some ultraproduct Q ?n(α)1 . As one interesting application of Section 2, in Section 3 they study the finite representability in {?n1 }n∈N in the system (N B(·, ·), νB). They obtain that C is the unique Banach space which is finitely representable in {?n1 }n∈N in the system (N B(·, ·), νB). This is quite strange and different from the classical result in Banach space theory.
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