A Finitely Additive State Criterion for the Completeness of Inner Product Spaces

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• 作者：Emmanuel Chetcuti and Anatolij Dvure?enskij
• 关键词：completion of inner product space ; finitely additive state ; Hilbert space ; inner product space (pre ; Hilbert space) ; orthogonally closed subspace
• 刊名：Letters in Mathematical Physics
• 出版年：2003
• 出版时间：2003
• 年：2003
• 卷：64
• 期：3
• 页码：221-227
• 全文大小：106 KB

文摘

We show that an inner product space S (real, complex or quaternion) is complete if, and only if, the system of all orthogonally closed subspaces in S, denoted by F(S), admits at least one finitely additive state which is not vanishing on the set of all finite dimensional subspaces of S. Although it gives only a partial solution to the problem formulated by Pták on the existence of a finitely additive state on F(S) for incomplete S, this gives an important insight into the structure of the set of states on F(S). This criterion has no analogue whatsoever in E(S), the system of splitting subspaces of S.