|
|
| |
| On the Kottman Problem |
| |
Citation： |
Yin Hongsheng.On the Kottman Problem[J].Chinese Annals of Mathematics B,1982,3(5):617~624 |
| Page view： 786
Net amount： 815 |
Authors： |
Yin Hongsheng; |
|
|
| Abstract： |
Let X be a real linear normed space of infinite dimension, and S ( X ) be the
unit sphere in X , i. e., S (X ) = {x\in X : ||x||=1}.
For any cardinal number \alpha, write
R_\alpha( X ) = sup {\lambda: there exists a subset A in S (X ) whose cardinal number is \alpha,such that for any two distinct elements x and y in A, || x—y || \geq \lambda holds}.
In this paper, we mainly discuss the relation between R_\alpha(X ) and R_\alpha(X^*), where X^* is the conjugate space of X .
We establish the following theorems.
Theorem 1. If R(X^*) < 2 , then R(X ) > 1 .
Theorem 4. R_3(X^*) =2 \Leftrightarrow R_3(X) =2.
Theorem 5. For any $b_1,b_2\in (1,2],b_1b_2 \geq 2$ , there exists an X such that
$R(X)=b_1,R(X^*)=b_2$.
In addition, Using F_\alpha(X) of Definition 4.1 in [2] (see Definiton 5 in this paper),
we prove the following theorem, which generalizes Theorem 4 .2 (b) in [2 ] ,
Theorem 3 . $R_\alpha(X^*)=2 \Leftrightarrow F_2(X)=0$. |
Keywords： |
|
Classification： |
|
|
|
Download PDF Full-Text
|
|
|
|
