2000年, 方习年在文献[1]中引入了$\omega$ -强凸空间的概念, 并得到了$\omega$ -强凸空间、$k$ -强凸空间和$(H)$性质之间的关系, 但未给出其对偶概念, 也未给出用切片刻画的特征.文献[2]注意到了文献[1]的不足之处, 引入了$\omega$ -强光滑空间的概念, 证明了$\omega$ -强凸性与$\omega$ -强光滑性具有对偶性质, 并讨论了$\omega$ -强光滑性与其它光滑性之间的关系, 用切片统一刻画了$\omega$ -强凸空间与$\omega$ -强光滑空间的特征.本文在文献[1-2]的基础上, 给出了$\omega$ -非常凸空间和$\omega$ -非常光滑空间的概念.以局部自反原理为工具, 证明了$\omega$ -非常凸空间和$\omega$ -非常光滑空间的对偶关系, 并给出了它们的特征刻画.此外, 用切片统一刻画了$\omega$ -非常凸空间与$\omega$-非常光滑空间的特征.

本文中, $X$表示Banach空间, $X^{\ast}$表示$X$的共轭空间. $X$和$X^{\ast}$的单位球面分别用$S\equiv S(X)=\{x: x\in X, \|x\| =1\}$和$S^{\ast}\equiv S(X^{\ast})$表示, 单位球分别用$U(X)=\{x: x\in X, \|x\|\leq1\}$和$U(X^{\ast})$表示.以Span$\{x_{1}, x_{2}, \cdots, x_{l}\}$表示$X$中$l$个元$x_{1}, x_{2}, \cdots, x_{l}$所生成的子空间; $\forall x\in S, $记$\sum(x)=\{x^{\ast}: x^{\ast}\in S^{\ast}, x^{\ast}(x) =\|x\|\};$ $\forall x^{\ast}\in S^{\ast}, $记

$A_{x^{\ast}}=\{x: x\in S, x^{\ast}(x)=\|x\|\};$

对$x\in S $, $\delta>0$, 用$F^{\ast}(x, \delta)$表示切片

$F^{\ast}(x, \delta)=\{x^{\ast}: x^{\ast}\in S^{\ast}, x^{\ast}(x)\geq1-\delta\} ;$

对$x^{\ast}\in S^{\ast}$, $\delta>0$, 用$F(x^{\ast}, \delta)$表示切片$F(x^{\ast}, \delta)=\{x: x\in S, x^{\ast}(x)\geq1-\delta\}$.

定义2.1  称$x_{0}\in S$是$X$的$\omega$ -非常(强)凸点, 若$\forall x^{\ast}\in\sum(x_{0}), \{x_{n}\}_{n=1}^\infty\subset S$, 使得$\forall k\in $N$, \lim x^{\ast}(x_{0} +x_{n_{1}}+\cdots+x_{n_{k}})=k+1 (n_{1}, \cdots, n_{k}\rightarrow\infty)$成立, 则$\{x_{n}\}_{n=1}^\infty$是相对弱(相对)紧集.若$\forall x_{0}\in S$都是$X$的$\omega$ -非常(强)凸点, 则称$X$是$\omega$ -非常(强)凸空间.

定义2.2  称$x\in S$是$X$的$\omega$ -非常(强)光滑点, 若$\forall x^{\ast}_{0}\in\sum(x)(\forall x^{\ast}_{0}\in S^{\ast} , x\in A_{x^{\ast}_{0}})$, $\{x^{\ast}_{n}\}_{n=1}^\infty\subset S$, 使得$\forall k\in $N$, \lim (x^{\ast}_{0} +x^{\ast}_{n_{1}}+\cdots+x^{\ast}_{n_{k}})(x)=k+1 (n_{1}, \cdots, n_{k}\rightarrow\infty)$成立, 则$\{x^{\ast}_{n}\}_{n=1}^\infty$是相对弱(相对)紧集.若$\forall x\in S$都是$X$的$\omega$ -非常(强)光滑点, 则称$X$是$\omega$ -非常(强)光滑空间.

显然, $\omega$ -强凸蕴含$\omega$ -非常凸, $\omega$ -强光滑蕴含$\omega$ -非常光滑.

定义2.3 ^[3] 称$X$为近非常凸(近强凸)的, 如果$\forall\{x_{n}\}_{n=1}^\infty\subset S$, 及某个$x^{\ast}\in\sum(x)$, 当$x^{\ast}(x_{n})\rightarrow 1(n\rightarrow\infty) $时, $\{x_{n}\}_{n=1}^\infty$是相对弱紧(相对紧).

定义2.4 ^[4] 称$X$具有$WS(S)$性质, 如果$\forall x\in S, \{x_{n}^{\ast}\}_{n=1}^\infty\subset S^{\ast}$, 当$x^{\ast}_{n}(x)\rightarrow 1(n\rightarrow\infty) $时, $\{x^{\ast}_{n}\}_{n=1}^\infty$是相对弱紧(相对紧).

由文献[5]中引理2易得

引理2.1  若$f_{1}, f_{2}, \cdots, f_{k}\in S^{\ast}$, $\varphi\in S^{\ast\ast}$且

$d(\varphi (f_{l}), \varphi ({\hbox{Span}} \{f_{1}, \cdots, f_{l-1}\})) \geq\epsilon, l=1, 2, \cdots, k, $

则

$d(\varphi (f_{l}), \varphi ({\hbox{Span}}\{f_{i}\}, i\neq l))\geq \frac{2\epsilon^{k}}{2^{k}}, l=1, \cdots, k.$

引理2.2  若$C$是$X$的弱紧子集, 使得$\forall f\in S^{\ast}$, $\epsilon>0$及一切自然数$n$有$d(f(x_{n}), f(C))<\epsilon$, 则存在$\{x_{n}\}$的子列$\{x_{n_{k}}\}$, 使得$\left | f(x_{n_{k}})-f(x_{n_{l}}) \right |<2\epsilon $.

证 据$d(f(x_{n}), f(C))<\epsilon $, 有$\{y_{n}\}\subset C$, 使$\left | f(x_{n})-f(y_{n}) \right |<\epsilon $.由$C$弱紧, $\{y_{n}\}$有弱收敛子列$\{y_{n_{k}}\}$, 不失一般性可设对, $k, l$, $\left | f(y_{n_{k}})-f(y_{n_{l}}) \right |<\epsilon $, 由此易知

$\left | f(x_{n_{k}})-f(x_{n_{l}}) \right |<2\epsilon.$

引理2.3 ^[3] $X$为近非常凸当且仅当$\sum \left ( x^{*} \right )=\hat{A}_{x^{*}}$.

引理2.4 ^[8] (局部自反原理) 设$X^{\ast}_{0}$和$X^{\ast\ast}_{0}$分别是$X^{\ast}$和$X^{\ast\ast}$的有限维子空间, 则对任何$\epsilon\in(0, 1)$, 存在--有界线性算子$T:X^{\ast\ast}_{0}\rightarrow X^{\ast}$, 使

(ⅰ) $(1-\epsilon )\left \| F \right \|\leq \left \| T(F) \right \|\leq (1+\epsilon )\left \| F \right \|;F\in X_{0}^{**};$

(ⅱ) $f(T(F))=F(f);f\in X_{0}^{*}, F\in X_{0}^{**};$

(ⅲ) $T(\hat{x})=x, x\in X\bigcap X_{0}^{**}. $

定理3.1  若$X^{\ast}$是$\omega$ -非常凸的, 则$X$是$\omega$ -非常光滑的.

证 设$x_{0}^{\ast}\in S^{\ast}, x\in A_{x_{0}^{\ast}}, \{x_{n}^{\ast}\}_{n=1}^\infty\subset S^{\ast}$且$\forall k\in $N$ $,

$\lim(x_{0}^{\ast} +x_{n_{1}}^{\ast}+\cdots+x_{n_{k}}^{\ast})(x)=k+1 (n_{1}, n_{2}, \cdots, n_{k}\rightarrow\infty).$

把$x$看成$S^{\ast\ast}$中的元, 则$\forall k\in $N$, \lim x(x_{0}^{\ast} +x_{n_{1}}^{\ast}+\cdots+x_{n_{k}}^{\ast})=k+1 (n_{1}, n_{2}, \cdots, n_{k} \rightarrow\infty)$成立, 并且由$x\in A_{x_{0}^{\ast}}$知, $x\in\sum(x_{0}^{\ast}).$再由$X^{\ast}$是$\omega$ -非常凸的假设知, $\{x_{n}^{\ast}\}_{n=1}^\infty$是相对弱紧集, 这说明$X$是$\omega$ -非常光滑的.

定理3.2  $X^{\ast}$是$\omega$ -非常光滑的当且仅当$X$是$\omega$ -非常凸且自反.

证 必要性.设$\forall x_{0}\in S, x^{\ast}\in\sum(x_{0}), \{x_{n}\}_{n=1}^\infty\subset S $, 且对$\forall k\in $N,

$ \lim x^{\ast}(x_{0} +x_{n_{1}}+\cdots+x_{n_{k}})=k+1 (n_{1}, n_{2}, \cdots, n_{k} \rightarrow\infty).$

把$x_{0}, x_{n}(n=1, \cdots, \infty)$看成$S^{\ast\ast}$中的元, 则由$ x^{\ast}\in\sum(x_{0})$知, $x^{\ast}(x_{0})=1$, 从而$x_{0}(x^{\ast})=1$, 故$x^{\ast}\in A_{x_{0}}$, 且$\forall k\in $N$, \lim (x_{0}+x_{n_{1}}+\cdots+x_{n_{k}})(x^{\ast})=k+1, (n_{1}, n_{2}, \cdots, n_{k}\rightarrow\infty)$成立.再由$X^{\ast}$是$\omega$ -非常光滑的假设知, $\{x_{n}\}_{n=1}^\infty$是相对弱紧集，这说明$X$是$\omega$ -非常凸的.

设$\forall x_{0}\in S, x^{\ast}\in\sum(x_{0})$, 则存在$\{x_{n}\}_{n=1}^\infty\subset S $, 使$x^{\ast}(x_{n})\rightarrow1$, 把$x_{0}, x_{n}(n=1, \cdots, \infty)$看成$S^{\ast\ast}$中的元, 则由$x^{\ast}\in\sum(x_{0})$和$x^{\ast}(x_{n})\rightarrow1$知, $x^{\ast}\in A_{x_{0}}, x_{n}(x^{\ast})\rightarrow1, $所以$\forall k\in $N$, \lim(x_{0}+x_{n_{1}}+\cdots+x_{n_{k}})(x^{\ast})=k+1 (n_{1}, n_{2}, \cdots, n_{k}\rightarrow\infty)$成立.再由$X^{\ast}$是$\omega$-非常光滑的假设知, $\{x_{n}\}_{n=1}^\infty$是相对弱紧集, 于是存在子列$\{x_{n_{j}}\}\subset\{x_{n}\}$, 使$\{x_{n_{j}}\}$弱收敛于$x^{\ast\ast}$, 其中$x^{\ast\ast}\in S^{\ast\ast}, x^{\ast\ast}(x^{\ast})=1$, 由Mazur定理知$J_{X}(X)$是弱闭的, 于是存在$x_{0}\in S, $使$\hat{x_{0}}=x^{\ast\ast}$, 因此$x^{\ast}(x_{0})=x^{\ast\ast}(x^{\ast})=1=\|x^{\ast}\|$, 由James定理知, $X$是自反的.

充分性.由假设条件和定理$3.1$立即得到.

与定理3.2类似地易证下面

定理3.3  (1) 若$X^{\ast}$是近非常凸的, 则$X$具有$WS$性质;

(2) $X^{\ast}$具有性质$(WS)$当且仅当$X$是近非常凸且自反.

定理3.4  (1) 若$X$是$\omega$ -非常凸的, 则$X$是近非常凸的.

(2) 若$X$是$\omega$ -非常光滑的, 则$X$是近非常光滑的.

证 (1) 如果$\forall x, \{x_{n}\}_{n=1}^\infty\subset S$, 及某个$x^{\ast}\in\sum(x)$, 有$x^{\ast}(x_{n})\rightarrow1(n\rightarrow \infty)$, 则必有$\lim x^{\ast}(x +x_{n_{1}}+\cdots+x_{n_{k}})=k+1 (n_{1}, n_{2}, \cdots, n_{k} \rightarrow\infty)$, 于是由$X$的$\omega$ -非常凸性知$\{x_{n}\}_{n=1}^\infty$是相对弱紧的, 这说明$X$是近非常凸的.

(2) 设$\forall\{x^{\ast}_{n}\}_{n=1}^\infty\subset S, x\in S$, 于是由Hahn-Banach定理知, 存在$x_{0}^{\ast}\in S^{\ast}$, 使$x_{0}^{*}(x)=\left \| x \right \|=1$, 故$x\in A_{x_{0}^{\ast}}$.因此, 当$x^{\ast}_{n}(x)\rightarrow1$时, $\forall k\in N$, 有

$\lim(x^{\ast}_{0}+x^{\ast}_{n_{1}}+\cdots+x^{\ast}_{n_{k}})(x)=k+1 (n_{1}, n_{2}, \cdots, n_{k}\rightarrow\infty), $

由$X$的$\omega$ -非常光滑性知$\{x^{\ast}_{n}\}_{n=1}^\infty$是相对弱紧的, 这说明$X$是近非常光滑的.

定理3.5  设$X$是自反空间, 则$\omega$ -非常凸空间当且仅当近非常凸.

证 定理3.4(1) 中已证必要性, 故下面只证充分性.如果$\forall x_{0}\in S, x^{\ast}\in\sum(x_{0}), \{x_{n}\}_{n=1}^\infty\subset S$, 且$\forall k\in N$, $\lim x^{\ast}(x_{0}+x_{n_{1}}+\cdots+x_{n_{k}})=k+1 (n_{1}, n_{2}, \cdots, n_{k}\rightarrow\infty)$.因$X$自反的, 所以有子列$\{x_{n_{j}}\}$弱收敛于$y$, 故$x^{^{*}}(y)=1, y\in S$, 即$\{x_{n}\}$为相对弱紧集, 进而$X$是$\omega$ -非常凸的.

由定理3.2--3.5得

推论3.1  若$X^{*}$是$\omega$ -非常光滑的, 则$X$是近非常凸且自反.

推论3.2  $X^{*}$是$\omega$ -非常光滑的当且仅当$X^{*}$具有$(WS)$性质.

定理3.6  $X$是$\omega$ -非常光滑的当且仅当$\forall x^{*}\in S^{*}$及$\epsilon >0, x\in A_{x^{\ast}}$, 存在$\delta =\delta (x^{*}, \epsilon )>0$及紧集$C^{*}\subset X^{*}$, 使得$\forall \varphi \in S^{**}$, 有$F^{\ast}(x, \delta)\subset\{y: y\in X^{\ast}, d( \varphi (y), \varphi (C^{\ast}))<\epsilon\}$.

证 必要性. 若存在$x_{0}^{\ast}\in S^{\ast}$及$\epsilon_{0}>0, x_{0}\in A_{x_{0}^{\ast}}, $对$\forall n\in $N$ $及任意紧集$C^{\ast}\subset X^{\ast}$, 存在$\phi _{0}\in S^{**}, $使$F^{\ast}(x_{0}, \frac{1}{n})$不包含在$\{y: y\in X^{\ast}, d(\phi _{0}(y), \phi _{0}(C^{\ast}))<\epsilon_{0}\}$中.由$x_{0}\in A_{x_{0}^{\ast}}$知, $x_{0}^{\ast}\in F^{\ast}(x_{0}, \frac{1}{n})$.记$C_{1}^{n}=\{y: y\in $Span$\{x_{0}^{\ast}\};\|y\|\leq3\}$, 则$C_{1}^{n}$是紧集, 因而$F^{\ast}(x_{0}, \frac{1}{n})$不包含在$\{y: y\in X^{\ast}, d(\phi _{0}(y), \phi _{0}(C_{1}^{n}))<\epsilon_{0}\}$中, 故可取$x_{1}^{\ast}\in F^{\ast}(x_{0}, \frac{1}{n})$, 使

$d(\phi _{0}(x_{1}^{\ast}), \phi _{0}({\hbox{Span}}\{x_{0}^{\ast}\})=d(\phi _{0}(x_{1}^{\ast}), \phi _{0}(C_{1}^{n}))\geq\epsilon_{0}. $

记$C_{2}^{n}=\{y: y\in $Span$ \{x_{0}^{\ast}, x_{1}^{\ast}\};\|y\|\leq3\}$, 则$C_{2}^{n}$仍是紧集, 因而$F^{\ast}(x_{0}, \frac{1}{n})$仍不包含在$\{y: y\in X^{\ast}, d(\phi _{0}(y), \phi _{0}(C_{2}^{n}))<\epsilon_{0}\}$中, 故可取$x_{2}^{\ast}\in F^{\ast}(x_{0}, \frac{1}{n})$, 使

$d(\phi _{0}(x_{2}^{\ast}), \phi _{0}({\hbox{Span}}\{x_{0}^{\ast}, x_{1}^{\ast}\}))=d(\phi _{0}(x_{2}^{\ast}), \phi _{0}(C_{2}^{n}))\geq\epsilon_{0}.$

继续上述过程, 可取$x_{0}^{\ast}, x_{1}^{\ast}, \cdots, x_{n-1}^{\ast}\in F^{\ast}(x_{0}, \frac{1}{n})$.记

$C_{n}^{n}=\{y: y\in {\hbox{Span}}\{x_{0}^{\ast}, x_{1}^{\ast}, \cdots, x_{n-1}^{\ast}\};\|y\|\leq3\}, $

则$C_{n}^{n}$仍是紧集, 因而$F^{\ast}(x_{0}, \frac{1}{n})$仍不包含在$\{y: y\in X^{\ast}, d(\phi _{0}(y), \phi _{0}(C_{n}^{n}))<\epsilon_{0}\}$中, 故可取$x_{n}^{\ast}\in F^{\ast}(x_{0}, \frac{1}{n})$, 使

$d(\phi _{0}(x_{n}^{\ast}), \phi _{0}({\hbox{Span}}\{x_{0}^{\ast}, x_{1}^{\ast}, \cdots, x_{n-1}^{\ast}\}))=d(\phi _{0}(x_{n}^{\ast}), \phi _{0}(C_{n}^{n}))\geq\epsilon_{0}(\forall n\in {\hbox{N}}), $

这表明$\{x_{n}^{\ast}\}_{n=1}^\infty$不是相对弱紧集.

另一方面, 由于$x_{n}^{\ast}\in F^{\ast}(x_{0}, \frac{1}{n})$, 故$x_{n}^{\ast}(x_{0})\geq 1-\frac{1}{n}$, 进而有$x_{n}^{\ast}(x_{0})\rightarrow 1, (n\rightarrow\infty)$, 于是对$\forall k\in $N$, \{n_{i}\} \subset\{n\}, i=1, \cdots, k, \lim (x_{0}^{\ast} +x_{n_{1}}^{\ast}+\cdots+x_{n_{k}}^{\ast})(x_{0})=k+1 (n_{1}, \cdots, n_{k}\rightarrow\infty)$成立.再由$X$是$\omega$ -非常光滑的假设知, $\{x_{n}^{\ast}\}_{n=1}^\infty$是相对弱紧集, 这与$\{x_{n}^{\ast}\}_{n=1}^\infty$不是相对弱紧集相矛盾.

充分性.若$X$不是$\omega$ -非常光滑的, 则存在$x_{0}^{\ast}\in S^{\ast}, x_{0}\in A_{x_{0}^{\ast}}, \{x_{n}^{\ast}\}_{n=1}^\infty\subset S^{\ast}$且$\forall k\in $N,

$ \lim(x_{0}^{\ast} +x_{n_{1}}^{\ast}+\cdots+x_{n_{k}}^{\ast})(x_{0})=k+1 (n_{1}, \cdots, n_{k}\rightarrow\infty) , $

但$\{x_{n}^{\ast}\}_{n=1}^\infty$不是相对弱紧集.由$ \lim(x_{0}^{\ast} +x_{n_{1}}^{\ast}+\cdots+x_{n_{k}}^{\ast})(x_{0})=k+1 (n_{1}, \cdots, n_{k}\rightarrow\infty)$知, $\forall n\in $N$, $存在$ $N$_{1}$, 当$n_{1}, \cdots, n_{k}>$N$_{1}$时,

$(x_{0}^{\ast} +x_{n_{1}}^{\ast}+\cdots+x_{n_{k}}^{\ast})(x_{0})>k+1-\frac{1}{2n}, $

从而$x_{n_{i}}^{\ast}(x_{0})>1-\frac{1}{2n} (i=1, 2, \cdots, k)$.由于$x_{0}^{\ast}\in S^{\ast}, x_{0}\in A_{x_{0}^{\ast}}$, 故对$\epsilon=\frac{1}{2n}$, 存在$\delta_{1}=\delta_{1}(x_{0}^{\ast}, \frac{1}{2n})>0$及紧集$C^{\ast}\subset X^{\ast}$, 使得$\forall\phi \in S^{**}$有

$F^{\ast}(x, \delta_{1})\subset\{y: y\in X^{\ast}, d(\phi (y), \phi (C^{\ast}))<\frac{1}{2n}\}.$

令$\delta=\max\{\delta_{1}, \frac{1}{2n}\}$, 则$x_{n_{i}}^{\ast}(x_{0})\geq1-\delta, (i=1, 2, \cdots, k)$且$F^{\ast}(x_{0}, \delta)\subset F^{\ast}(x_{0}, \delta_{1})$, 因而$F^{\ast}(x_{0}, \delta)\subset\{y: y\in X^{\ast}, d(\phi (y), \phi (C^{\ast}))<\frac{1}{2n}\}$, 于是

$\{x_{n_{1}}^{\ast}\}, \{x_{n_{2}}^{\ast}\}, \cdots, \{x_{n_{k}}^{\ast}\}\subset F^{\ast}(x_{0}, \delta)\subset\{y: y\in X^{\ast}, d(\phi (y), \phi (C^{\ast}))<\frac{1}{2n}\}, $

由引理$2.2$, 存在子序列$\{x_{n_{k}^{'}}^{\ast}\}\subset\{x_{n_{k}}^{\ast}\}$, 使$\left | \phi (x_{n_{k}^{'}}^{\ast})-\phi (x_{n_{j}^{'}}^{\ast}) \right |<2\frac{1}{2n}=\frac{1}{n}$, 由$n\in $N$ $的任意性, 采用对角线法可得$\{x_{n_{k}}^{\ast}\}$的弱收敛子列, 这导致$\{x_{n}^{\ast}\}_{n=1}^\infty$是相对弱紧集, 这与$\{x_{n}^{\ast}\}_{n=1}^\infty$不是相对弱紧集相矛盾.

定理3.7  $X$是$\omega$ -非常凸的当且仅当$\forall x\in S $及$\epsilon>0, x^{\ast}\in\sum(x)$, 存在$\delta=\delta(x, \epsilon)>0$及紧集$C\subset X $, 使得$\forall f\in S^{*}$, 有$F(x^{\ast}, \delta)\subset\{y: y\in X, d(f(y), f(C))<\epsilon\}$.

证 必要性. 若存在$x_{0}\in S $及$\epsilon_{0}>0, x_{0}^{\ast}\in\sum(x_{0})$, $\forall n\in $N$ $及任意紧集$C\subset X $, 存在$f_{0}\in S^{*}$, 使$F(x_{0}^{\ast}, \frac{1}{n})$不包含在$\{y: y\in X, d(f_{0}(y), f_{0}(C))<\epsilon_{0}\}$中, 由$x_{0}^{\ast}\in\sum(x_{0})$知, $x_{0}\in F(x_{0}^{\ast}, \frac{1}{n})$.记$C_{1}^{n}=\{y: y\in $Span$\{x_{0}\};\|y\|\leq3\}$, 则$C_{1}^{n}$是紧集, 因而$F(x_{0}^{\ast}, \frac{1}{n})$不包含在$\{y: y\in X, d(f_{0}(y), f_{0}(C_{1}^{n}))<\epsilon_{0}\}$中, 故可取$x_{1}\in F(x_{0}^{\ast}, \frac{1}{n})$, 使

$d(f_{0}(x_{1}), f_{0}({\hbox{Span}} \{x_{0}\}))=d(f_{0}(x_{1}), f_{0}(C_{1}^{n}))\geq\epsilon_{0}. $

记$C_{2}^{n}=\{y: y\in $Span$\{x_{0}, x_{1}\};\|y\|\leq3\}$, 则$C_{2}^{n}$仍是紧集, 因而$F(x_{0}^{\ast}, \frac{1}{n})$仍不包含在$\{y: y\in X, d(f_{0}(y), f_{0}(C_{2}^{n}))<\epsilon_{0}\}$中, 故可取$x_{2}\in F(x_{0}^{\ast}, \frac{1}{n})$, 使

$d(f_{0}(x_{2}), f_{0}({\hbox{Span}}\{x_{0}, x_{1}\}))=d(f_{0}(x_{2}), f_{0}(C_{2}^{n}))\geq\epsilon_{0}. $

继续上述过程, 可取$x_{0}, x_{1}, \cdots, x_{n-1}\in F(x_{0}^{\ast}, \frac{1}{n})$, 记

$C_{n}^{n}=\{y: y\in {\hbox{Span}}\{x_{0}, x_{1}, \cdots, x_{n-1}\};\|y\|\leq3\}, $

则$C_{n}^{n}$仍是紧集, 因而$F(x_{0}^{\ast}, \frac{1}{n})$仍不包含在$\{y: y\in X, d(f_{0}(y), f_{0}(C_{n}^{n}))<\epsilon_{0}\}$, 故可取$x_{n}\in F(x_{0}^{\ast}, \frac{1}{n})$, 使

$d(f_{0}(x_{n}), f_{0}({\hbox{Span}}\{x_{0}, x_{1}, \cdots, x_{n-1}\}))=d(f_{0}(x_{n}), f_{0}(C_{n}^{n}))\geq\epsilon_{0}(\forall n\in {\hbox{N}}). $

这表明$\{x_{n}\}_{n=1}^\infty$不是相对弱紧集.

另一方面, 由于$x_{n}\in F(x_{0}^{\ast}, \frac{1}{n})$, 故$x_{0}^{\ast}(x_{n})\geq1-\frac{1}{n}$, 进而有$x_{0}^{\ast}(x_{n})\rightarrow 1 (n\rightarrow\infty)$, 于是$\forall k\in $N$, \{n_{i}\}\subset\{n\}, i=1, \cdots, k, \lim x_{0}^{\ast}(x_{0} +x_{n_{1}}+\cdots+x_{n_{k}})=k+1 (n_{1}, \cdots, n_{k}\rightarrow\infty)$成立, 再由$X$是$\omega$ -非常凸的假设知, $\{x_{n}\}_{n=1}^\infty$是相对弱紧集, 这与$\{x_{n}\}_{n=1}^\infty$不是相对弱紧集相矛盾.

充分性.  若$X$不是$\omega$ -非常凸的, 则存在$x_{0}\in S , x_{0}^{\ast}\in\sum(x_{0}), \{x_{n}\}_{n=1}^\infty\subset S $且$\forall k\in $N$, \lim x_{0}^{\ast}(x_{0} +x_{n_{1}}+\cdots+x_{n_{k}})=k+1 (n_{1}, \cdots, n_{k}\rightarrow\infty)$, 但$\{x_{n}\}_{n=1}^\infty$不是相对弱紧集.由$\lim x_{0}^{\ast}(x_{0} +x_{n_{1}}+\cdots+x_{n_{k}})=k+1 (n_{1}, \cdots, n_{k}\rightarrow\infty)$知, $ \forall n\in $N$, $存在$ $N$_{1}$, 当$n_{1}, \cdots, n_{k}>$N$_{1}$时, $x_{0}^{\ast}(x_{0} +x_{n_{1}}+\cdots+x_{n_{k}})>k+1-\frac{1}{2n}$, 从而$x_{0}^{\ast}(x_{n_{i}})>1-\frac{1}{2n} (i=1, 2, \cdots, k)$.由于$x_{0}\in S, x_{0}^{\ast}\in\sum(x_{0})$, 故对$\epsilon=\frac{1}{2n}$, 存在$\delta_{1}=\delta_{1}(x_{0}, \frac{1}{2n})>0$及紧集$C\subset X$, 使得$\forall f\in S^{*}$, 有

$F(x_{0}^{\ast}, \delta_{1})\subset\{y: y\in X, d(f(y), f(C))<\frac{1}{2n}\}.$

令$\delta=\max\{\delta_{1}, \frac{1}{2n}\}$, 则$x_{0}^{\ast}(x_{n_{i}})\geq1-\delta (i=1, 2, \cdots, k)$且$F(x_{0}^{\ast}, \delta)\subset F(x_{0}^{\ast}, \delta_{1})$, 因而$F(x_{0}^{\ast}, \delta)\subset\{y: y\in X, d(f(y), f(C))<\frac{1}{2n}\}$, 于是

$\{x_{n_{1}}\}, \{x_{n_{2}}\}, \cdots, \{x_{n_{k}}\}\subset F(x_{0}^{\ast}, \delta)\subset\{y: y\in X, d(f(y), f(C))<\frac{1}{2n}\}, $

由引理$2.2$, 存在子序列$\{x_{n_{k}^{'}}\}\subset\{x_{n_{k}}\}$, 使$\left | f(x_{n_{k}^{'}})-f(x_{n_{j}^{'}})\right |<2\frac{1}{2n}=\frac{1}{n}$, 由$n\in N $的任意性, 采用对角线法可得$\{x_{n_{k}}\}$的弱收敛子列, 这导致$\{x_{n}\}_{n=1}^\infty$是相对弱紧集, 这与$\{x_{n}\}_{n=1}^\infty$不是相对弱紧集相矛盾.

根据局部自反原理, 可得到$\omega$ -非常凸空间和$\omega$ -非常光滑空间之间更深刻的对偶性质.

定理3.8  (1) $X$是$\omega$ -非常凸空间当且仅当$\forall x\in S(X), x^{\ast}\in\sum(x)$是$X^{\ast}$的$\omega$ -非常光滑点;

(2) $X$是$\omega$ -非常光滑空间当且仅当$\forall x\in S(X), x^{\ast}\in\sum(x)$是$X^{\ast}$的$\omega$ -非常凸点.

证 (1) 充分性.如果$\forall x\in S, x^{\ast}\in\sum(x), \{x_{n}\}_{n=1}^\infty\subset S$, 使得$\forall k\in N$, 有

$\lim x^{\ast}(x+x_{n_{1}}+\cdots+x_{n_{k}})=k+1 (n_{1}, n_{2}, \cdots, n_{k}\rightarrow\infty)$

成立, 因$x^{\ast}\in\sum(x)(\hat{x}\in A_{x^{*}})$是$X^{\ast}$的$\omega$ -非常光滑点, 把$x$和$\{x_{n}\}_{n=1}^\infty$是看成$S^{**}$中的点列, 则$\lim (x+x_{n_{1}}+\cdots+x_{n_{k}})(x^{\ast})=k+1 (n_{1}, n_{2}, \cdots, n_{k}\rightarrow\infty)$成立, 于是根据已知条件推出$\{x_{n}\}_{n=1}^\infty$是相对弱紧的, 故$X$是$\omega$ -非常凸空间.

必要性. 若存在$x_{0}\in S(X), $使$x_{0}^{\ast}\in\sum(x_{0})$不是$X^{\ast}$的$\omega$ -非常光滑点, 则存在$x_{0}^{\ast\ast}\in\sum(x_{0}^{\ast})$和$\{x_{n}^{\ast\ast}\}_{n=1}^\infty\subset S^{\ast\ast}$, 使

$\lim (x_{0}^{\ast\ast}+x_{n_{1}}^{\ast\ast}+\cdots+x_{n_{k}}^{\ast\ast})(x_{0}^{\ast})=k+1 (n_{1}, n_{2}, \cdots, n_{k}\rightarrow\infty)$

成立, 但$\{x_{n}^{\ast\ast}\}_{n=1}^\infty$不是相对弱紧集, 从而$\{x_{n}^{\ast\ast}\}_{n=1}^\infty$的任何子列$\{x_{m}^{\ast\ast}\}_{m=1}^\infty$均不弱收敛, 故$\forall y^{\ast\ast}\in U(X^{\ast\ast})$, $\{x_{m}^{\ast\ast}\}_{m=1}^\infty$不弱收敛于$y^{\ast\ast}$.因此, 存在$\{y_{m}^{\ast}\}_{m=1}^\infty\subset S^{\ast}$, 使得$\{x_{m}^{\ast\ast}\}_{m=1}^\infty$不弱收敛于$y^{\ast\ast}$, 即存在$\epsilon_{0}>0, \{y_{m}^{\ast}\}_{m=1}^\infty\subset S^{\ast}$, 使得

$\left | x_{m}^{\ast\ast}(y_{m}^{\ast})-y^{\ast\ast}(y_{m}^{\ast})\right |\geq \epsilon_{0}. $

(★)

因$X$是$\omega$ -非常凸空间, 由定理3.4知$X$是近非常凸的, 再由引理2.3知$\sum(x_{0}^{\ast})=\hat{A}_{x_{0}^{\ast}}$, 因此存在$x^{0}\in A_{x_{0}^{\ast}}, $使得$x^{0}=x_{0}^{\ast\ast}$.

任取$\{x_{m}^{\ast\ast}\}_{m=1}^\infty$的$k$个子列$\{x_{m_{i}(m)}^{\ast\ast}\}_{m=1}^\infty, i=1, 2, \cdots, k$, 其中$m_{i}(m)$表示$\{m_{i}\}$的第$m$项.对每个$m$, 令$X_{m}^{\ast\ast}=$Span$\{x^{0}, x_{m}^{\ast\ast}, x_{m_{1}(m)}^{\ast\ast}, \cdots, x_{m_{k}(m)}^{\ast\ast}\} , X_{m}^{\ast}=$Span$\{x_{0}^{\ast}, y_{m}^{\ast}\}$, 由局部自反原理, 存在一个--有界线性算子$T_{m}:X_{m}^{\ast\ast}\rightarrow X$满足:

(ⅰ) $ 1-\frac{1}{m}\leq \left \| T_{m}(x_{m}^{**}) \right \|\leq 1+\frac{1}{m};1-\frac{1}{m}\leq \left \| T_{m}(x_{m_{i}(m)}^{**}) \right \|\leq 1+\frac{1}{m} (i=1, 2, \cdots, k)$;

(ⅱ) $ x_{0}^{*}(T_{m}(x_{m}^{**}))=x_{m}^{**}(x_{0}^{**});y_{m}^{*}(T_{m}(x_{m_{i}(m)}^{**}))=x_{m_{i}(m)}^{**}(y_{m}^{*}) (i=1, 2, \cdots , k)$;

(ⅲ) $ T_{m}(x^{0})=x^{0}$.

令$x_{m}=\frac{T_{m}(x_{m}^{**})}{\left \| T_{m}(x_{m}^{**}) \right \|}, x_{m_{i}(m)}=\frac{T_{m}(x_{m_{i}(m)}^{**})}{\left \| T_{m}(x_{m_{i}(m)}^{**}) \right \|}(i=1, 2, \cdots, k)$, 显然$\{x_{m}\}_{m=1}^\infty\subset S$且$\{x_{m_{i}(m)}\}$是$\{x_{m}\}$的子列, $i=1, 2, \cdots, k$, 由(ⅱ)得

$x_{0}^{*}(x_{m_{i}(m)})=\frac{x_{0}^{*}(T_{m}(x_{m_{i}(m)}^{**}))}{\left \| T_{m}(x_{m_{i}(m)}^{**}) \right \|}=\frac{x_{m_{i}(m)}^{**}(x_{0}^{*})}{\left \| T_{m}(x_{m_{i}(m)}^{**}) \right \|}(i=1, 2, \cdots, k).$

因此$\lim x_{0}^{\ast}(x^{0}+x_{m_{1}(m)}+\cdots+x_{m_{k}(m)})=k+1 (m_{1}, m_{2}, \cdots, m_{k}\rightarrow\infty)$.已知$X$是$\omega$ -非常凸空间, 所以$\{x_{m}\}$是相对弱紧集.设$\{x_{m_{k}}\}$是$\{x_{m}\}$的收敛子列, 则$\{x_{m_{k}}\}$弱收敛于某个$x$.在($\star$)中取$y^{**}=x$, 则由($\star$)和(ⅱ)得

$\lim\left | y_{m_{k}}^{\ast}(x_{m_{k}})-y_{m_{k}}^{\ast}(x)\right | =\lim\left | y_{m_{k}}^{\ast\ast}(y_{m_{k}}^{*})-x(y_{m_{k}}^{\ast})\right |\geq \epsilon_{0}, $

这与$\{x_{m_{k}}\}$弱收敛于$x$相矛盾.

与(1) 对偶地易证(2), 故证明从略.

与定理3.8类似地, 能够给出$\omega$ -强凸空间和$\omega$ -强光滑空间之间更深刻的对偶性质, 进而可推广文献[2]中给出对偶定理.

定理3.9  (1) $X$是$\omega$ -强凸空间当且仅当$\forall x\in S(X), x^{\ast}\in\sum(x)$是$X^{\ast}$的$\omega$ -强光滑点;

(2) $X$是$\omega$ -强光滑空间当且仅当$\forall x\in S(X), x^{\ast}\in\sum(x)$是$X^{\ast}$的$\omega$ -强凸点.