| THE FUNCTIONAL DIMENSION OF SOME CLASSES OF SPACES |
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| 作者姓名： | LIU Shangping LI Bingren |
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| 作者单位： | LIU SHANGPING LI BINGREN Institute of Mathematics，Academy of Mathematics and System Sciences，Chinese Academy of Sciences，Beijing 100080，China. Institute of Mathematics，Academy of Mathematics and System Sciences，Chinese Academy of Sciences，Beijing 100080，China |
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| 基金项目： | Project supported by the National Natural Science Foundation of China (No.10071088, No.10171098). |
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| 摘 要： | 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.
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| 关 键 词： | 可数希尔伯特空间 拓扑线性空间 函数维度 巴拿赫空间 计算公式 |
| 收稿时间： | 2013/11/3 0:00:00 |
THE FUNCTIONAL DIMENSION OF
SOME CLASSES OF SPACES |
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| LIU Shangping,LI Bingren.THE FUNCTIONAL DIMENSION OF SOME CLASSES OF SPACES[J].Chinese Annals of Mathematics,Series B,2005,26(1):67-74. |
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| Authors: | LIU Shangping and LI Bingren |
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| Institution: | Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences,Beijing,100080, China |
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| Abstract: | 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. |
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| Keywords: | Functional dimension Countable Hilbert space Topological linear space |
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