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$l^{0}(\{X_{i}\})$型赋$F$-范空间的共轭空间的表示定理
引用本文:王见勇.$l^{0}(\{X_{i}\})$型赋$F$-范空间的共轭空间的表示定理[J].数学研究及应用,2017,37(5):591-602.
作者姓名:王见勇
作者单位:常熟理工学院数学系, 江苏 常熟 215500
基金项目:国家自然科学基金(Grant No.11471236).
摘    要:作者在《数学学报》(2016, {\bf 59}(4))上的一篇文章中, 给出了几个$l^{0}$型赋$F$-范空间的共轭空间的表示定理. 对于赋范空间序列$\{X_{i}\}$,本文研究$l^{0}(\{X_{i}\})$型赋$F$- 范空间的共轭空间的表示问题,得到代数表示连等式$\left(l^{0}(\{X_{i}\})\right)^{*}\stackrel{A}{=}\left(c^{0}_{00}(\{X_{i}\})\right)^{*}\stackrel{A}{=}c_{00}(\{X^{*}_{i}\})$,$$\left(l^{0}(X)\right)^{*}\stackrel{\mathrm{A}}{=}\left(c^{0}(X)\right)^{*}\stackrel{\mathrm{A}}{=}\left(c_{0}^{0}(X)\right)^{*}\stackrel{\mathrm{A}}{=}\left(c^{0}_{00}(X)\right)^{*}\stackrel{\mathrm{A}}{=}c_{00}(X^{*}),$$以及序列弱星拓扑下的拓扑表示定理$\left(c^{0}_{00}(\{X^{*}_{i}\}),sw^{*}\right)=c^{0}_{00}(\{X^{*}_{i}\})$. 对于内积空间序列与通常拓扑下的数域空间序列,文章最后给出了基本表示定理的具体表现形式.

关 键 词:$l^{0}(\{X_{i}\})$型赋$F$-范空间    局部凸空间    局部有界空间    序列弱星拓扑    表示定理
收稿时间:2016/8/11 0:00:00
修稿时间:2016/11/11 0:00:00

The Representation Theorems of Conjugate Spaces of Some $l^{0}(\{X_{i}\})$ Type $F$-Normed Spaces
Jianyong WANG.The Representation Theorems of Conjugate Spaces of Some $l^{0}(\{X_{i}\})$ Type $F$-Normed Spaces[J].Journal of Mathematical Research with Applications,2017,37(5):591-602.
Authors:Jianyong WANG
Institution:Department of Mathematics, Changshu Institute of Technology, Jiangsu 215500, P. R. China
Abstract:In a paper published in {\sl Acta Mathematica Sinica} (2016, 59(4)) we obtained some representation theorems for the conjugate spaces of some $l^{0}$ type $F$-normed spaces. In this paper, for a sequence of normed spaces $\{X_{i}\}$, we study the representation problems of conjugate spaces of some $l^{0}(\{X_{i}\})$ type $F$-normed spaces, obtain the algebraic representation continued equalities $$(l^{0}(\{X_{i}\}))^{*}\stackrel{A}{=}(c_{00}^{0}(\{X_{i}\}))^{*}\stackrel{A}{=}c_{00}(\{X^{*}_{i}\}),$$ $$(l^{0}(X))^{*}\stackrel{\mathrm{A}}{=}(c^{0}(X))^{*}\stackrel{\mathrm{A}}{=}(c_{0}^{0}(X))^{*}\stackrel{\mathrm{A}}{=}(c_{00}^{0}(X))^{*}\stackrel{\mathrm{A}}{=}c_{00}(X^{*}),$$ and the topological representation $((c^{0}_{00}(\{X_{i}\}))^{*},sw^{*})=c^{0}_{00}(\{X^{*}_{i}\})$, where $sw^{*}$ is the sequential weak star topology. For the sequences of inner product spaces and number fields with the usual topology, the concrete forms of the basic representation theorems are obtained at last.
Keywords:$l^{0}(\{X_{i}\})$ type $F$-normed space  locally convex space  locally bounded space  sequential weak star topology  representation theorem
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