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| THE FUNCTIONAL DIMENSION OFSOME CLASSES OF SPACES |
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Citation： |
LIU Shangping,LI Bingren.THE FUNCTIONAL DIMENSION OFSOME CLASSES OF SPACES[J].Chinese Annals of Mathematics B,2005,26(1):67~74 |
| Page view： 1026
Net amount： 874 |
Authors： |
LIU Shangping; LI Bingren |
Foundation： |
Project supported by the National Natural Science Foundation of China (No.10071088, No.10171098). |
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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 $\sigma(B^*, B)$. Then the functional dimension of
$(B^*,\sigma(B^*, B))$ is 1. By the third result, a class of
topological linear spaces with finite functional dimension is
presented. |
Keywords： |
Functional dimension, Countable Hilbert space, Topological linear space |
Classification： |
46A50, 46B20 |
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