THE FUNCTIONAL DIMENSION OF SOME CLASSES OF SPACES

• Published in: Chinese annals of mathematics. Serie B Vol. 26; no. 1; pp. 67 - 74
• Main Authors: LIU, SHANGPING, LI, BINGREN
• Format: Journal Article
• Language: English
• Published: Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences,Beijing,100080, China 2005
• Subjects: 函数维度; 可数希尔伯特空间; 巴拿赫空间; 拓扑线性空间; 计算公式; Countable Hilbert space; Topological linear space; Functional dimension
• ISSN: 0252-9599; 1860-6261
• DOI: 10.1142/s0252959905000063

The functional dimension of countable Hilbert spaces has been discussed by some authors. They showed that every countable Hilbert space with finite functional dimension is nuclear. In this paper the authors do further research on the functional dimension, and obtain the following results: (1) They construct a countable Hilbert space, which is nuclear, but its functional dimension is infinite. (2) The functional dimension of a Banach space is finite if and only if this space is finite dimensional. (3)Let B be a Banach space, B^* be its dual, and denote the weak * topology of B^* by σ(B^*, B)， Then the functional dimension of (B^*, σ(B^*, B)) is 1. By the third result, a class of topological linear spaces with finite functional dimension is presented.