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关于希尔伯特空间的特徵性质
ON A CHARACTERIZATION OF HILBERT SPACES

刘良深;
LAU LEUNG-SUM

中山大学,
(Chung-Shan University

收稿日期: 1958-09-25
出版日期: 1958-08-15

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摘要  在巴拿赫空间B中引入内积这一问题首先由P.Jordan及J.V.Neumann所解决。其后研究这个问题的有Kakutani,Lorch,Mackey,Day,Kasahara等诸氏:我们在这里利用了共轭同构(Conjugate isomorphism)的概念把Lorch在1945年的工作简化。现在把我们所需要的概念和定义叙述如下。定义设B为巴拿赫空间,B 为B之共轭空间,若B上之元素能与B上之元素建
Abstract:It was well known that Hilbert space is self-adjoint space in the sense of conjugate isomorphism. The converse problem was studied by Lorch and a sufficient condition for a self-conjugate space to be a Hilbert space was obtained. The present note gives a necessary and sufficient condition for this problem and in turn simplifies the result of Lorch. In fact, the following theorems will be proved: Theorem 1. Let B be a complex Banach space, the necessary and sufficient condition for B to be a complex Hilbert space is 1) B(?)B~* (B is conjugate isomorphic to B~*), 2) f_x(x)=||x||~2 for every x in B,where f_x is the corresponding functional of x under the isomorphism. Theorem 2. Let B be a real Banach space, the necessary and sufficient condition for B to be a real Hilbert space is 1) B=B~*(B is self-adjoint), 2) f_x(x)=||x||~2 for every x in B, where f_x is the corresponding functional of x under the isomorphism.
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