$l^{0}(\{X_{i}\})$型赋$F$-范空间的共轭空间的表示定理
The Representation Theorems of Conjugate Spaces of Some $l^{0}(\{X_{i}\})$ Type $F$-Normed Spaces

DOI：10.3770/j.issn:2095-2651.2017.05.009

 作者 单位 王见勇 常熟理工学院数学系, 江苏 常熟 215500

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.

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.